Notes on algebraic stacks
Fredrik Meyer
Abstract
These are lecture notes from a course on algebraic stacks I attended
during the fall semester 2013. They follow very closely the lectures,
which in turn followed a draft version of the book “Algebraic spaces
and stacks” by Martin Olsson. The lectures were held by Paul Arne
Østvær. All mistakes are my own.
Contents
1 Moduli problems, spaces, and stacks. Vector
K-theory
1.1 Some category theory . . . . . . . . . . . . . .
1.2 Back to moduli spaces . . . . . . . . . . . . . .
1.3 The way out of the problem . . . . . . . . . . .
1.4 Algebraic stacks and moduli of vector bundles .
1.5 K-theory of schemes . . . . . . . . . . . . . . .
2 Flatness, limits of schemes and Kähler
2.1 Limits of schemes . . . . . . . . . . . .
2.2 Kähler differentials . . . . . . . . . . .
2.3 Kähler differentials on schemes . . . .
bundles and
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3
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differentials
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3 Étale maps
23
3.1 Étale and smooth maps . . . . . . . . . . . . . . . . . . . . . 23
3.2 Invariance of étale sites under infinetesimal thickenings . . . . 27
4 Functors, Hilbert Polynomial, Grassmannians
4.1 The Hilbert polynomial . . . . . . . . . . . . .
4.2 The Grassmannian . . . . . . . . . . . . . . . .
4.3 The Grassmannian as a functor . . . . . . . . .
4.4 Quot and Hilbert schemes . . . . . . . . . . . .
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28
30
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35
36
5 Topologies, sites, sheaves
38
5.1 Grothendieck topologies and sites . . . . . . . . . . . . . . . . 38
5.2 Presheaves and sheaves . . . . . . . . . . . . . . . . . . . . . . 40
6 Fibered categories
47
6.1 Fibered categories . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 2-Yoneda-lemma . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 Categories fibered in Set . . . . . . . . . . . . . . . . . . . . . 52
7 Groupoids and descent
54
7.1 Groupoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.2 Fiber products of categories fibered in groupoids . . . . . . . 56
7.3 Descent theory . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8 Descent theorems
8.1 Descent for sheaves on a site . . . .
8.2 Descent for quasi-coherent sheaves
8.3 Descent for closed subschemes . . .
8.4 Descent for open embeddings . . .
8.5 Descent for affine morphisms . . .
8.6 Descent for quasiaffine morphisms .
8.7 Descent for polarized schemes . . .
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61
61
62
66
66
67
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68
9 Example, torsors, principal homogeneous spaces
69
9.1 The moduli stack of curves of genus g ≥ 2 . . . . . . . . . . . 69
9.2 Torsors / principal homogeneous spaces . . . . . . . . . . . . 69
10 Stacks and algebraic spaces
10.1 Definition of stacks . . . . . . . . .
10.2 Algebraic spaces . . . . . . . . . .
10.3 Algebraic spaces as sheaf quotients
10.4 Examples of algebraic spaces . . .
10.5 More on algebraic spaces . . . . . .
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74
74
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78
78
11 Algebraic stacks
81
11.1 First results on algebraic stacks . . . . . . . . . . . . . . . . . 81
11.2 Properties of Artin stacks . . . . . . . . . . . . . . . . . . . . 84
11.3 The moduli stack of curves of genus ≥ 2 . . . . . . . . . . . . 87
2
12 Root stacks and the Weil conjecures
89
12.1 Root stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
12.2 The Weil conjectures . . . . . . . . . . . . . . . . . . . . . . . 92
A Exam
94
B References to the litterature
94
B.1 Background litterature . . . . . . . . . . . . . . . . . . . . . . 94
1
Moduli problems, spaces, and stacks. Vector bundles and K-theory
We all know of varieties. In this course we will learn of wide generalizations
of varieties. Here’s some containments: varieties ⊂ schemes ⊂ algebraic
spaces ⊂ Deligne-Mumford stacks ’69 ⊂ Artin stacks ’74.
So, at the end of this course, you should be able to answer the question,
when stopped on the street, “what’s an algebraic stack?”.
Algebraic stacks were introduced by Deligne in his SGA4 and by Giroud
in his thesis [[source needed]].
The need for stacks arose in the study of moduli spaces in algebraic
geometry. Roughly, a moduli space is a geometric space (scheme or stack)
whose points represents algebro-geometric data of some kind, or isomorphism
classes of such. Some examples:
• RP1 is the moduli space of lines through the origin in R2 .
• The Grassmannian G(n, V ) is the moduli space of all n-dimensional
linear subspaces of the vector space V .
Then, what is a moduli problem?
Definition 1.1. A moduli problem is a contravariant functor
F : Schop → Set.
We say that a scheme M ∈ Sch is a fine moduli space for F if it represents
F, that is, if
F(X) ≈ Hom(X, M )
for all schemes X.
3
1.1
Some category theory
Here, C is a category and X ∈ ob(C). We define the functor hX : C op →
Set. On objects: Y 7→ C(Y, X), and on morphisms: by composing. One
checks that this defines a contravariant functor. We then say that a functor
F : C op → Set is representable if there exists X ∈ ob(C) with F ≈ hX
(isomorphism of functors).
g
Now, let X →
− Y be a morphism in C. This induces a morphism h(g) :
hX → hY by sending a morphism f to the composition g ◦ f ∈ hY (Z). Given
h(g), we can recover g, as the image of idX ∈ hX (X) under h(g)(X).
More generally, suppose we are given a morphism of functors (or a natural
transformation, same thing) η : hX → F. This gives us a distinguished
element τη ∈ F(X), that is the image of idX ∈ hX (X) under ηX . In fact,
this is a bijection.
Lemma 1.2 (Yoneda’s lemma). There’s a bijection between natural transformations
η : hX → F
and the set F(X), and it is given by η 7→ τη .
Example 1.3. When F is represented by Y , that is F = hY , we get a
bijection
HomC (X, Y ) ≈ Nat(hX , hY )
F
Exercise 1. Prove Yoneda’s lemma, plus the following statement: If F :
≈
C op → Set is given, then a pair (X, η), X ∈ ob(C) and η : hX −
→ F is unique
up to unique isomorphism.
♠
♥
Solution 1.
1.2
Back to moduli spaces
The category of affine schemes AffSch is equivalent to the category CRop ,
the opposite category of the category of commutative rings. Here are some
moduli problems for commutative rings.
Example 1.4. Let F : CR → Set be defined by sending R to R× , that is,
sending a ring to its set of units. Then one can check that Z[x, x1 ] is a fine
moduli space for units, because HomCR (Z[x, x1 ], R) ∼
F
= R× .
4
Example 1.5. Let CR → Set be defined by sending a ring R to the isomorphism class of (M ; m1 , . . . , mn ), where M is a locally free R-module of rank
r generated by m1 , . . . , mr . Two such pairs are isomorphic if there exists
an isomorphism α with α(mi ) = m0i . On maps, one just sends a ring map
R → S to the map of modules M 7→ M ⊗R S.
There is a fine moduli space for this problem as well. It is the Grassmannian. G(r, n) = Proj(Z[xI ]/Ir,n ), where Ir,s is the Plücker ideal.
F
Example 1.6. Fix a numerical polynomial P (z) ∈ Q[z]. Recall that a
numerical polynomial is a polynomial that for large enough integer values of
Z, outputs only integers. Define the moduli functor HilbertP,n : Schop → Set
by sending a scheme X to the set of all schemes Y ⊆ PnX = PnZ ×Z X flat
over X and such that the fibres of x ∈ X all have Hilbert polynomial P . We
define what happens to morphisms by base change.
Then there is a theorem of Grothendieck, that there exists a fine moduli
space for HilbertP,n , called the Hilbert scheme. It is projective, and in fact
contained in a Grassmannian.
F
Example 1.7.
Hilb1 (Pn ) = Pn
Hilbn (Pn ) = Pn
The last equality because Pn is the n’th symmetric product of P1 .
F
However, a fine moduli space doesn’t always exist!
Definition 1.8. We say that X ∈ Schk is a coarse moduli space for a moduli
problem F : Schop → Set if the following two conditions hold:
1. There exists a universal morphism ρX : F → hX :
F
ρX
f
/ hX
∃!hf
hY
2. For every algebraically closed field k̄ ⊃ k, the morphism
ρX (k̄) : F(k̄) → hX (k̄)
is a bijection.
5
Remark. Fine implies coarse.
Now for a rather lengthy example to motivate why coarse moduli spaces
are necessary. Define the moduli functor Fell : Schop
Q → Set by sending X to
isomorphism classes of elliptic curves over X.
We recall what an elliptic curve is. In general, an elliptic curve over S is
an S-scheme together with a map p : E → S satisfying the three conditions
below:
• p is proper and smooth of relative dimension 1.
• The geometric fibers of p are connected curves of genus 1.
• There exists a section 0 ∈ E(s) = HomSch (S, E).
The first condition is explained in Hartshorne, chapter 3, paragraph ten. In
particular, it means that for all s ∈ S, the fiber p−1 (s) = E ×S Spec(k(s)) =
¯
Es is geometrically regular over k(s), i.e. the base change of Es to Spec(k(s))
is non-singular.
The second condition means that the curve Ek defined by the pullback
Ek
/E
p
Spec(k)
/S
has arithmetic genus 1, meaning that dimk H 1 (Ek , O Ek ) = 1.
Claim: The moduli problem Fell has no fine moduli space. This is
seen like this: the two curves defined by the equations y 2 = x3 − x and
2y 2 = x3 − x√are non-isomorphic over Q, but become isomorphic over Q̄ (via
(x, y) 7→ (x, 2y)). This implies that Fell (Q) 6,→ Fell (Q̄) is not an injection,
but injectivity would have held if Fell were representable.
Question: Is the obstruction for a fine moduli space for Fell trivial if we
work over an algebraically closed field? Turns out: NO!
The reason is because elliptic curves have non-trivial automorphisms.
Because of the equation y 2 = f (x), we see that every elliptic curve has at
least one non-trivial automorphism.
Suppose Mell is a fine moduli space for Fell . Then Fell (Mell ) = HomSch (Mell , Mell ),
which contains the special element idMell , which corresponds to “universal
elliptic curve” Euniv → Mell . Then one checks that with these definitions,
6
every elliptic curve is the pullback via a unique map X → Mell :
E
/ Euniv
/ Mell
X
But if E admits automorphisms, this map is not unique! (why?)
Luckily, there is a way forward by realizing that moduli problems live in
a “2-category of fibered categories over Sch”.
A stack is what you get by starting with a groupoid and adding geometry.
A groupoid is a category where all morphisms are isomorphisms.
1.3
The way out of the problem
Let Gpd be the collection of groupoids. That is, it consists of categories in
which every morphism is an isomorphism.
Example 1.9. Every set S can be regarded as a groupoid by setting
(
idX if X = Y
Hom(X, Y ) =
∅
otherwise
In this way we can identify Set as a full subcategory of Gpd.
We want to redefine a moduli problem as a (pseudo-)functor
F : Schop → Gpd.
If we regard hX (Y ) as a groupoid, then every such object has a trivial automorphism group.
F
Thus we need to abandon the category of schemes. We’ll replace it by
the “2-category of stacks” which accomodates automorphisms.
1.4
Algebraic stacks and moduli of vector bundles
In this course we will study the moduli functor
VBundlesnX : Schop
k → Set
given by sending a scheme Y to the set of isomorphism classes of rank n
vector bundles over X ×Z Y . On maps, base change.
7
Example 1.10. Now for a rather lengthy example. We will construct an
example of a rank 2 vector bundle on P2k . Cover P2k with the standard affine
patches, U0 , U1 , U2 with affine coordinates ( xx01 , xx20 ) on U0 etc.
We want to construct the transition functions ρ10 : U0 ∩ U1 → U0 ∩ U1 .
Let λ ∈ k × . Then we have the relation:
  
 
x1 2
x1
x0
0
( x0 )
 x0  = 
  x1 
x2
x2
(1 − λ) xx1 x2 2 λ xx10
x0
x1
0
The entries in the 2 × 2-matrix are regular functions on U0 ∩ U1 . Similarly,
there is a relation ρ21 : U1 ∩ U2 → U1 ∩ U2 :
  
 
0 x2 (1 − λ0 ) x0 x2
x0
x0
λ
x21   x2 
 x1  =  x1
x22
x2
x1
0
x1
x2
x2
1
Combining these two transition maps gives us the transition map ρ20 =
ρ21 ◦ ρ10 :
  
 
0 x1 x2
0 ) x2
x1
x0
λ
(1
−
λ
x0
x20
x2 
 x0  = 


2
2
x
x
x2
x1
λ0 (1 − λ) x22 (λλ0 − λ0 + 1) x0 x2 1
x0
x2
0
To avoid division of zero on U2 ∩ U0 we must thus have λλ0 − λ0 + 1 = 0,
1
that is, λ0 = 1−λ
. Hence if λ 6= 0, 1, we can glue the patches Ui × A2k . Map
2
F
to Pk via the evident projection. This gives a vector bundle over P2k .
For convenience we repeat the definition of a vector bundle:
Definition 1.11. Let X be a scheme over a field k. Then a vector bundle
over X is a Zariski trivial map π : E → X. That is, there exists an open
'
covering of X and isomorphisms ϕi : π −1 (Ui ) −
→ Ui × Ank such that for all i, j
there exists transition maps ϕij : Ui ∩Uj → GLn (k) such that ϕi ◦ϕ−1
j (x, v) =
(x, ϕij (x)v) for all x ∈ Ui ∩ Uj and v ∈ Ank . We call the number n the rank
of E.
The next proposition is homework:
Proposition 1.12. There is an equivalence between the category VBundlesk
of vector bundles of rank n over X and the category of locally free sheaves of
rank n on X.
Proof. Also see [Neu09], Chapter I.
Also this: Exercise II.5.18 in Hartshorne and exercise 6.10 and 6.11 in
Hartshorne, the latter two are about K-theory of schemes.
8
1.5
K-theory of schemes
We introduce some of the main characters of K-theory of schemes. Here we
assume that X is a Noetherian scheme.
Let Vb(X) denote the set of isomorphism classes of vector bundles of
finite rank on X. Let Coh(X) denote the set of isomorphism classes of
coherent sheaves on X.
Now, define K00 (X) = Z{Coh(X)}/ ∼, that is the free abelian group
on the set Coh(X) under the following equivalence relation: for every exact
sequence
0 → F 0 → F → F 00 → 0
we identify [F] with [F 0 ] + [F 00 ]. Similarly, we define K0 (X) via Vb(X).
Lemma 1.13. For K0 and K00 , the following is true.
f
• Any scheme map X −
→ Y induces a map on K-theory:
f ∗ : K0 (Y ) → K0 (X)
given by [E] 7→ [f ∗ E].
• A flat map f : X → Y induces a map f ∗ : K00 (Y ) → K00 (X), defined
in the same way as the above map.
• Tensor products of locally free sheaves turn K0 (X) into a ring and
K00 (X) into a K0 (X)-module.
• If f : X → Y is a proper map, then it induces a pushforward map
f∗ : K 0 (X) → K00 (Y ) given by
f∗ ([F]) =
∞
X
(−1)i [Ri f∗ F],
i=0
where the Ri f∗ F are the higher derived images of F.
• If f : X → Y is proper and α ∈ K00 (X) and β ∈ K0 (Y ), we have a
projection formula:
f∗ (f ∗ (β) · α) = β · f∗ (α).
Here f ∗ (β) ∈ K0 (X) and f∗ (α) ∈ K00 (Y ), the equality taking place in
K00 (Y ).
9
Proof. The first three statements are obvious from the defintions.
By standard scheme theory (for example Hartshorne III, section 8.8)
there is an induces long exact sequence of coherent sheaves:
0 → f∗ F 0 → f∗ F → f∗ F 00 → R1 f∗ F 0 → · · · → Rn f∗ F 00 → 0
The statement now follows by breaking this long exact sequence into short
exact sequences.
The last statement follows from the formula
Ri f∗ (f ∗ (E) ⊗O X F) = E ⊗O Y Ri f∗ (F),
which is Exercise 8.3 in Chapter III in Hartshorne.
Here’s a remarkable theorem:
Theorem 1.14. If X is regular, the two K-theories coincide, that is, we
have an isomorphism
≈
K0 (X) −
→ K00 (X).
The theorem says that even though one category (the category of coherent sheaves) is much larger than the other category (the category of vector
bundles), their K-theories coincide.
For more on K-theory, see for example [Ros94].
10
2
Flatness, limits of schemes and Kähler differentials
We start by recalling definitions of flatness. Let R be a ring and M and
R-module. Then T (−) = (−) ⊗R M is an endofunctor of ModR . We say that
M is flat over R if T (−) is an exact functor.
Definition 2.1. We say that an R-module M is faithfully flat if for all
R-modules N and N 0 the induced homomorphism
HomR (N, N 0 ) → HomR (N ⊗R M, N 0 ⊗R M )
is injective.
We have a long list of equivalences:
Proposition 2.2. The following are equivalent:
1. M is faithfully flat.
2. M is flat and for all R-modules N 0 , the map
N 0 → HomR (M, N 0 ⊗R M )
y 7→ (m 7→ y ⊗ m)
is injective.
3. A sequence of R-modules
E : N 0 → N → N 00
is exact if and only if
E ⊗ M : N 0 ⊗R M → N ⊗R M → N 00 ⊗R M
is exact.
4. A homomorphism N ,→ N 0 is injective if and only if
N 0 ⊗R M ,→ N ⊗R M
is injective.
5. M is flat and N ⊗R M = 0 implies N = 0.
11
6. M is flat and for all maximal ideals m ∈ max(R), we have M/mM 6= 0.
Proof. 1 ⇔ 2 : Assume 1). Then put N = R in the definition of faithfully
flat. This implies 2). For the other direction, let F = ⊕i∈I R be some free
R-module surjecting onto N . Then we have a commutative diagram:
0
HomR (N,
 N )
/ HomR (N ⊗R M, N 0 ⊗R M )
_
_
/ HomR (F ⊗R M, N 0 ⊗ M )
HomR (F, N 0 )
Q
Q
But HomR (F, N 0 ) ' i∈I N 0 and HomR (F ⊗R M, N 0 ⊗R M ) ' i∈I HomR (M, N 0 ⊗R
M ), and so by assumption, the bottom row is injective. Then by commutativity, the top arrow is injective, hence M is faithfully flat.
2 ⇒ 4 : One direction is obvious, so assume N 0 ⊗R M ,→ N ⊗R M is
injective. We have a commutative diagram:
N0

/ HomR (M, N 0 ⊗R M )
/ HomR (M, N ⊗R M )

N
If we assume the right arrow is injective, it follows by commutativity that
the left arrow is commutative also.
4 ⇒ 5 : Put N = 0.
5 ⇒ 6 : Put N = R/m.
5 ⇔ 3 : 3) can be rephrased as “M is flat and for any short exact sequence
E (as above), the sequence E ⊗ M is also exact”. This implies 5) by looking
at 0 → N → 0.
We want to show 3) using 5). Let H = ker(N → N 00 )/ im(N 0 → N ).
Tensoring with M and using flatness, we get
H ⊗R M =
ker(N ⊗R → N 00 ⊗R M )
im(N 0 ⊗R M → N ⊗R M )
The bottom sequence E ⊗ M in 3) is exact if and only if H ⊗R M = 0. By
5), this implies H = 0, which implies that E is exact.
It remains to show 6 ⇒ 2. Assume that M is flat, but that the map in 2)
is not injective. So let x ∈ N 0 be some non-zero element such that its image
under N 0 → HomR (M, N 0 ⊗R M ) is zero, that is, the map x 7→ x ⊗ m is zero
·x
for all x. Let ρx : R −
→ N 0 be multiplication by x and let a = ker ρx , so we
have an exact sequence of R-modules:
ρx
0 → a → R −→ N 0 .
12
Tensoring this with M we get the exact sequence
ρx ⊗id
0 → aM → M −−−−→ N 0 ⊗R M.
But the last map is the zero map, so by exactness aM = M . Let m be some
maximal ideal containing a. Then also mM = M , so M/mM = 0, so 6) is
wrong.
f
Definition 2.3. We say that a morphism of schemes X −
→ Y is flat if
#
f : O Y,f (x) → O X,x is flat for all x ∈ X. We say that f is faithfully flat if
it is flat and surjective.
Remark. Let R → R0 be a ring map. Then the map of affine schemes
Spec R0 → Spec R is flat if and only if R0 is a flat R-module. The same for
faithfully flat.
To see this: flatness is a local property, so the first statement is trivial.
For the second statement, assume the map of affine schemes is not surjective.
Then there is some prime ideal p ∈ Spec R such that no q in R0 maps to p
as the inverse image of f # . Then f # (p) is a unit in R0 for all p ∈ p. This
implies that the map
0 → p ⊗R R0 → R0
p ⊗ r 7→ f # (p)r
is surjective, and so R0 is not faithfully flat over R. The argument can
be reversed.
f
Proposition 2.4. La X −
→ Y be a flat map between locally noetherian
schemes of locally finite type. Then f is an open map.
Proof. This is Theorem 2.12 in [Mil80].
Here is an important corollary that will be used repeatedly in the next
lectures.
Corollary 2.5. Let f be as above. Let {Ui } be an affine cover of Y . Then
for every index i there exists a Zariski open cover {Vij } of f −1 (Ui ) where Vij
is quasi-compact and f (Vij ) = Ui .
Proof. Let p ∈ f −1 (Ui ) and let Wip ⊆ f −1 (Ui ) be an open affine neighbourhood of p. By the proposition, the image f (Wip ) is open in Y , and it is
contained in the affine set Ui which is quasi-compact. Letting p vary, we
13
see that Ui = ∪p f (Wip ), and by quasi-compactness, we can choose a finite
number of points pj such
S that Ui = ∪pj f (Wipj ).
Set Vip = Wip ∪ q Wipq . Then Vip is a finite union of quasi-compact
sets, so is itself quasi-compact, and clearly f (Vip ) = Ui .
Letting p vary, we have exhibited an open cover {Vip } of f −1 (Ui ) for
every i, satisfying the conclusion in the statement.
Definition 2.6. Let A be a ring and M and A-module. We say that M is
finitely presented if there is an exact sequence
Ar → As → M → 0
for some natural numbers r, s1 . Note that if A is Noetherian then this is
equivalent to A being just finitely generated.
f
If A −
→ B is a ring map, we say that B is a finitely presented A-algebra
if there is a surjection
π : A[x1 , . . . , xn ] → B
with finitely generated kernel. Again: if A is noetherian, this is equivalent to
B being just a finitely generated A-algebra (by Hilbert’s basis theorem). These definitions have of course their analogues in the category of schemes
and sheaves:
Definition 2.7. We say that a quasi-coherent sheaf F on a scheme X is
locally finitely presented if for all open affines Spec(B) ⊆ X, the B-module
Γ(Spec(B), F) is a finitely presented B-module.
f
We say that a map X −
→ Y of schemes is locally of finite presentation if for
each affine open set Spec B of Y , and affine cover f −1 (Spec B) = ∪ Spec Ai of
the inverse image, each ring map B → Ai is finitely presented as B-algebras.
We say that the morphism is of finite presentation if it is quasi-compact and
quasi-separated.
Remark. Recall that a morphism f : X → Y is quasi-separated if for every
affine open subset U of Y , the inverse image f −1 (U ) is a quasi-separated
scheme. Recall further that this means that the intersection of any quasicompact open sets are quasicompact. For more on numerous properties of
schemes, see Ravi Vakil’s ever-developing notes [Vak13].
Lemma 2.8. Locally finitely presented maps are closed under composition.
1
Zero is a natural number.
14
Proof. This follows from the corresponding facts of ring maps: If A → B
(resp. B → C) are finitely presented A-algebras (resp. B-algebras), then C
is a finitely presented A-algebra. This is easy to see.
Here’s a useful observation:
f
Lemma 2.9. Let X −
→ Y be a locally finitely presented faithfully flat map
(fppf ). Let E be a sheaf on X. Then the following is true: f∗ f ∗ E ≈ E.
Proof. Both are sheaves on X, so it is enough to compare stalks. We have:
(f∗ f ∗ E)x := lim (f∗ (f ∗ E))(U )
U 3x
= lim (f ∗ E)(f −1 (U ))
U 3x
= lim
lim
U 3x V ⊇f (f −1 (U ))
E(V )
= lim E(U ) = Ex
U 3x
The last equality follows because f is surjective and open.
2.1
Limits of schemes
Fix a partially ordered set (I, ≥). Assume that I is filtered, meaning that
for any pair of elements, there is some element that is greater than both of
them. We can think of I as a category, in which there is an arrow λ → µ if
and only if λ ≥ µ.
Fix a scheme B. A projective system/inverse system of B-schemes, indexed by I, is a functor S. : I → SchB . That is, for all λ ∈ I, we have a
B-scheme Sλ . For all λ ≥ µ, some µ ∈ I, we have maps
θλµ : Sλ → Sµ .
We say that S. have affine transition maps if each θλµ is an affine map.
Example 2.10. Let B = Spec Z and A a ring. Let I consist of all finitely
generated subrings of A and order them by inclusion. We get a projective
system in which each Sλ = Spec(Aλ ). All the transition maps are inclusion
maps.
F
Lemma 2.11. Let S. be a projective system of B-schemes with affine transition maps. Then the inverse limit limλ∈I Sλ exists in the category of Bschemes. For all λ ∈ I, the map
lim Sλ → Sλ
λ∈I
is affine.
15
“Proof ”. Since I is directed, we can assume I has a least element λ0 , since
the direct limit is unaffected by this. Then for every λ, the map Sλ → Sλ0
is affine. Then for each of these we have a quasi-coherent sheaf of algebras,
Aλ := θλλ0 ∗ O Sλ0 , and we can form the relative spectrum of this, we have
equalities Sλ = Spec(Aλ ) for all λ. Now, the inverse limit exists in the
category of sheaves, and so A = colimλ∈I Aλ exists. Define S = Spec(A).
We get maps ρλ : S → Sλ and properties of Spec shows that S = limλ∈I Sλ .
f
Proposition 2.12 (EGA, IV 8.14.2). A map X −
→ Y is of locally finite
presentation if and only if for every projective system of Y -schemes {Sλ }λ∈I ,
where each Sλ is affine, the map
colimλ∈I HomY (Sλ , X) → HomY (lim Sλ , X)
λ∈I
is a bijection.
Proof. This is Proposition 31.5.1 in [Sta13, Tag 01ZB].
Remark. The bijection in the proposition can be more compactly written as
colim hX (Sλ ) = hX (lim Sλ ).
Example 2.13. Let Y = Spec k. Let X = Spec A, where A is a k-algebra,
not finitely generated over k. Let I be the set of finitely generated subalgebras of A, ordered by inclusion. Then X = lim Sλ , where Sλ = Spec Aλ ,
λ ∈ I. Then idX ∈ HomY (limλ∈I Sλ , X) = HomY (X, X) is not in the image
of colimλ∈I homY (Sλ , X).
F
2.2
Kähler differentials
Let A → B be a ring map. Let M a B-module. An A-derivation of B is an
A-linear map, satisfying the Leibniz rule. That is, for all f, g ∈ B, we have
d(f g) = f d(g) + gd(f ).
This implies that d(a) = 0 for all a ∈ A. We denote by DerA (B, M ) the set
of all such derivations.
Definition 2.14. Let B be an A-algebra. The module of Kähler differentials
of B over A is the B-module Ω1B/A with an A-derivation d : B → Ω1B/A
16
satisfying the universal property:
B
d0
/M
O
!
d
∃!
Ω1B/A
Given d0 a derivation, it factors uniquely through Ω1B/A .
Proposition 2.15. The pair (Ω1B/A , d) exists and is unique up to unique
isomorphism.
Proof. Standard hocus pocus.
By the universal property, we have an identification:
≈
HomB (Ω1B/A , M ) −
→ DerA (B, M ).
So in fancy language, the module of differentials is represented by the module
of Kähler differentials (or corepresented, depending on your taste).
Exercise 2. Let B be an A-module, and let ρ : B ⊗A B → B be the diagonal
map, b1 ⊗ b2 7→ b1 b2 . View B ⊗A B as a B-module via b · (b1 ⊗ b2 ) = b1 ⊗ bb2 .
Let I = ker ρ. Then also I/I 2 is a B-module. Define
d : B → I/I 2
b 7→ db = b ⊗ 1 − 1 ⊗ b.
Then d is an A-derivation and I/I 2 satisfies the universal property of Ω1B/A .
So we have an isomorphism (Ω1B/A , d) ≈ (I/I 2 , d).
♠
Solution 2. That I/I 2 is a B-module is obvious. We check that d : B →
I/I 2 is a A-derivation. Let f, g ∈ B. First note that
(1 ⊗ f − f ⊗ 1)(1 ⊗ g − g ⊗ 1) = 1 ⊗ f g − g ⊗ f − f ⊗ g + f g ⊗ 1
is equal to zero modulo I 2 . This is the same as saying
1 ⊗ fg + fg ⊗ 1 = f ⊗ g + g ⊗ f
modulo I 2 . Thus:
f d(g) + gd(f ) = (1 ⊗ f g − g ⊗ f ) + (1 ⊗ f g − f ⊗ g)
= 1 ⊗ fg − 1 ⊗ fg + 1 ⊗ fg − fg ⊗ 1
= 1 ⊗ fg − fg ⊗ 1
= d(f g)
17
(1)
That d(A) = 0 follows by setting g = 1 and f = a for a ∈ A and applying
the just proven Leibniz rule.
It remains to prove the universal property. We have a canonical map
h : Ω1B/A → I/I 2 defined by df = 1⊗f −f ⊗1. First off, it is a surjection, since
I is generated as an ideal by exactly the expressions of the form 1 ⊗ fP− f ⊗ 1.
Too see this: clearly all such
P expressions liePI. Conversely,
P suppose
P fi ⊗ gi
is
in
I.
This
means
that
f
g
=
0
so
that
f
⊗g
=
f
⊗g
+
fi gi ⊗1 =
i
i
i
i
i
i
P
fi (1 ⊗ gi + gi ⊗ 1).
So the map h : Ω1B/A → I/I 2 is surjective. We now show that there is
a map s : I/I 2 → Ω1B/A such that s ◦ f = id, proving that f is injective as
well. Namely, define s : I/I 2 → Ω1B/A by f ⊗ g 7→ f dg. There is at least
such a map from I by properties of the tensor product and the bilinearity
of (f, g) 7→ f dg. Now, let 1 ⊗ f − f ⊗ 1 be a generator of I, and resp. for
1 ⊗ g − f ⊗ g. Their product is as in in (1) and is mapped to
d(f g) − gd(f ) − f d(g) + f gd(1).
This is mapped to zero by the Leibniz rule. Hence we have a well-defined
map s : I/I 2 → Ω1B/A . Also:
s ◦ h(df ) = s(1 ⊗ f − f ⊗ 1) = df − f d(1) = df.
Thus h is a bijection, and we have established the universal property of
(I/I 2 , d).
♥
Example 2.16. Let B = A[T1 , . . . , Tn ] be the polynomial ring in n variables. Then Ω1B/A is the free B-module generated by the symbols dTi for
i = 1, . . . , n.
F
Example 2.17. If B is a localization or a quotient of A then Ω1B/A = 0. F
ρ
Let B −
→ C be a map of A-algebras. In particular, we have a map
γ : A → B. Then we have natural maps of C-modules:
α : Ω1B/A ⊗A C → Ω1C/A
db ⊗ c 7→ cd(ρ(b))
and
β : Ω1C/A → Ω1C/B
adc 7→ γ(a)dc
Then we have:
18
Proposition 2.18. Let B be an A-algebra. Then the following holds:
1. Base change. For A0 an A-algebra, let B 0 = B ⊗A A0 . Then there is
an isomorphism of B 0 -modules Ω1B 0 /A0 ≈ Ω1B/A ⊗B B 0 .
2. If B → C is a map of A-algebras with α, β as above, then the sequence
α
β
→ Ω1C/A −
→ Ω1C/B → 0
Ω1B/A ⊗B C −
is exact.
3. Localization. If S is a multiplicatively closed subset of B, then
S −1 Ω1B/A ≈ Ω1S −1 B/A .
4. Conormal sequence. If C = B/I, we have an exact sequence
δ
α
I/I 2 →
− Ω1B/A ⊗B C −
→ Ω1C/A → 0
where δ(b) = db ⊗ 1.
Proof. For 1), the derivation d : B → Ω1B/A induces an A0 -derivation
d0 := d ⊗ idA0 : B 0 → Ω1B/A ⊗A A0 .
Now use the fact that Ω1B/A ⊗A A0 = Ω1B/A ⊗B B 0 , and check that the pair
(Ω1B/A ⊗B B 0 , d0 ) satisfies the universal property. Or else just compute an
explicit isomorphism.
Number 2) is Proposition 16.2 in Eisenbud [Eis95]. Number 3) is Proposition 16.9. Number 4) is Proposition 16.3.
Corollary 2.19. Let B be a finitely generated A-algebra, or a localization
of one. Then also Ω1B/A is finitely generated over B.
Proof. This follows from 4) in Proposition 2.18 and Example 2.16.
Example 2.20. Let B = A[T1 , . . . , Tn ] og let F ∈ B and let C = B/(F ).
Then
Ω1C/A = ⊕ni=1 CdTi /CdF
P
∂F
where dF = ni=1 ∂T
dTi .
F
i
Exercise 3. Let A be a ring. Show the two following statements:
19
1. Let B1 og B2 be A-algebras and R = B1 ⊗A B2 . Then
ϕ : (Ω1B1 /A ⊗B1 R) ⊕ (Ω1B2 /A ⊗B2 R) → Ω1R/A
is an isomorphism, where ϕ(db1 ⊗r1 +db2 ⊗r2 ) = r1 d(b1 ⊗1)+r2 d(b2 ⊗1).
So “Ω turns products into sums”.
2. Let B be an A-algebra and set C = B[T1 , . . . , Tn ]/I for some ideal
I. If α and δ is defined as above, show that there is a surjection of
B-modules ker δ → ker α.
♠
Solution 3. We prove that it is an isomorphism by giving an inverse. A
general element of Ω1R/A is a sum of elements of the form d(b1 ⊗ b2 ), where
b1 ∈ B1 and b2 ∈ B2 . Let ψ(b1 ⊗ b2 ) = d(b1 ) ⊗ (1 ⊗ b2 ) + d(b2 ) ⊗ (b1 ⊗ 1). I
claim that this is an inverse of ϕ.
Write r1 = f1 ⊗ f2 for a pure tensor in R. Such tensors generate R. We
first check that ψ ◦ ϕ = id:
ψ ◦ ϕ(db1 ⊗ (f1 ⊗ f2 ) + db2 ⊗ (g1 ⊗ g2 ))
= ψ((f1 ⊗ f2 )d(b1 ⊗ 1) + (g1 ⊗ g2 )d(1 ⊗ b2 ))
= (f1 ⊗ f2 )(db1 ⊗ (1 ⊗ 1)) + (g1 ⊗ g2 )(db2 ⊗ (1 ⊗ 1))
= db1 ⊗ (f1 ⊗ f2 ) + db2 ⊗ (g1 ⊗ g2 )
For the equality on the second line, we used that d(1) = 0, and the third line
used that R-multiplication in Ω1Bi /A ⊗ Bi R is done on the second factor. So
ψ ◦ ϕ = id.
We check that ϕ ◦ ψ = id, using the same notation as above:
ϕ ◦ ψ(d(b1 ⊗ b2 )) = ϕ(db1 ⊗ (1 ⊗ b2 ) + db2 ⊗ (b1 ⊗ 1))
= (1 ⊗ b2 )d(b1 ⊗ 1) + (b1 ⊗ 1)d(1 ⊗ b2 )
= d((b1 ⊗ 1)(1 ⊗ b2 )) = d(b1 ⊗ b2 )
The last line used the Leibniz rule. The proposition is proved.
♥
Here is a lemma that will be useful later when we talk about étale maps.
Lemma 2.21. Let k be a field and E a field extension of k. Put K =
E[T ]/(P (t)) for a polynomial P . Then if E is separable over k, the Kähler
'
differentials vanish: Ω1K/E = 0 and Ω1E/k ⊗E K −
→ Ω1K/k is an isomorphism.
20
Proof. Let P 0 (t) be the derivative of P . Then Ω1K/E = Kdt/P 0 (t)dt '
E[T ]/(P 0 (T ), P (T )). Separability implies that P (t) and P 0 (t) have no common factor, hence they generate the unit ideal, hence ΩK/E = 0.
2.3
Kähler differentials on schemes
The “obvious” construction yields what we want. By abuse of notation, we
also denote the resulting sheaf by Ω1X/Y .
f
Proposition 2.22. Let X −
→ Y be a map of schemes. Then there exists a
unique quasi-coherent sheaf Ω1X/Y on X such that for any open affine V ⊆ Y
and U ⊆ f −1 (V ) and x ∈ U , we have Ω1X/Y |U ' Ω1O X^
(U )/ O Y (U ) . At each
stalk, we have
(Ω1X/Y )x ' Ω1O X,x / O Y,f (x) .
Proof. This is a standard construction. We define Ω1X/Y (U ) to be the set of
functions
a
s:U →
ΩO X,x / O Y,f (x)
x∈U
such that for every x ∈ U , there exists an affine open neighbourhood Vy ,
where y = f (x) and an open affine Ux ⊆ f −1 (Vy ) of x and an ω in ΩO X (Ux )/ O Y (Vy )
such that ωx0 = s(x0 ) for all x0 ∈ Ux .
Then one sees immediately that Ω1X/Y is a sheaf of O X -modules, and
that the statements in the proposition holds.
Another, more coordinate-independent way to define Ω1X/Y is the following: let ∆ : X → X ×Y X be the diagonal map and let I = ker(O X×Y X →
∆∗ O X ) be the kernel of the associated map of sheaves. Then it turns out
that Ω1X/Y ' ∆∗ (I/I 2 ). This holds essentially by Exercise 2. See [Har77,
Chapter II, §8] for details.
Example 2.23. For any scheme Y , let X = AnY . Then Ω1X/Y ≈ O nX .
F
Example 2.24. Let A be a ring and X = P1A . Then ΩX/ Spec A ≈ O X (−2).
Compare with [Har77, Chapter II, §8], Thm 8.13. Why? We compute that
on the open set D+ (x0 ) ∩ D+ (x1 ) we have
x20 d(
x1
x0
) = −x21 d( ).
x0
x1
The coordinate on D+ (x0 ) can be identified with X := xx10 and the coordinate
on D+ (x1 ) with Y := xx01 . Then the calculation above says just that on the
21
intersection, we have dX = X12 d( X1 ), but applying the transition map X 7→
1
2
1
X = Y , this says dX = Y dY , so O X ' O X (2) ⊗ ΩX/ Spec A . Untwisting
gives Ω1P1 /A = O P1 (−2).
F
The following properties holds for Kähler differentials on schemes. Note
that they are completely analogous to the results for Kähler differentials of
modules from the previous lecture.
f
Proposition 2.25. Let X −
→ Y be a scheme map. Then:
1. Base change: If Y 0 is a Y -scheme, let X 0 = X ⊗Y Y 0 . Then
Ω1X 0 /Y 0 = p∗ Ω1X/Y ,
where p : X 0 → X is the natural projection.
2. Exact sequence: Let Y → Z be a scheme map. Then there is an exact
sequence of sheaves on X:
f ∗ Ω1Y /Z → Ω1X/Z → Ω1X/Y → 0.
3. Conormal sequence: Let j : Z ,→ X be a closed subscheme defined by
a quasi-coherent sheaf of ideals I. Then there is an exact sequence:
j ∗ (I/I 2 ) → ΩX/Y ⊗ O Z → ΩZ/Y → 0
of sheaves on Z.
The proofs follows directly from Proposition 2.18 by definition of Ω1X/Y .
22
3
Étale maps
Étale maps are like local homeomorphisms in topology. In particular, they
are what let us define the fundamental group of an algebraic variety.
3.1
Étale and smooth maps
We define étale and smooth maps of schemes. Compare with Section 10 in
Chapter III in [Har77].
f
Definition 3.1. A map of schemes X −
→ Y is formally smooth (resp. formally unramified /formally étale) if for every affine Y -scheme Y 0 and every
closed embedding j : Y00 ,→ Y 0 defined by a nilpotent ideal, the map
HomY (Y 0 , X) → HomY (Y00 , X)
f 7→ f ◦ j
is surjective (resp. injective/bijective).
If in addition f is of locally finite presentation, then f is called smooth
(resp. unramified /étale).
Remark. Assume Y00 → Y 0 is defined by I ⊆ O Y 0 with I n = 0 for some
n. Set Yi0 ⊆ Y 0 to be the closed subscheme defined by I i+1 . This gives us a
series of injections:
0
Y00 ,→ Y10 ,→ . . . ,→ Yn−1
=Y0
0
with Yi0 defined in Yi+1
by a square-zero ideal.
0
HomY (Y0 , X) factors as
Then HomY (Y 0 , X) →
0
HomY (Y 0 , X) → HomY (Yn−2
, X) → · · · → HomY (Y00 , X),
so in the definition, we could replace the word “nilpotent” by “square zero”.
Let S be a set and G a group acting on it. Then we say that S is a
G-torsor if the action of G is transitive.
Consider the set of liftings (dotted arrows) of y0 in the diagram below:
Y 00
y0
_
Y0
23
/X
?
Proposition 3.2. The set of dotted liftings as above is either empty or a
torsor under the action of the group HomY00 (y0∗ Ω1X/Y , I).
Proof. After some translation, I think this is just a global version of Lemma
4.5 in [Har10].
Here are some properties that are easy to check:
Proposition 3.3. The following holds:
f
1. If X −
→ Y is smooth (resp. unramified/étale) and g : Y 0 → Y any
scheme map, then f 0 : X ×Y Y 0 → Y 0 is smooth (resp. unramified/étale).
2. Smooth/unramified/étale maps are preserved under composition.
3. Suppose given a composite of maps that are locally of finite presentation
f
g
X−
→Y →
− Z with gf and g smooth and such that f ∗ Ω1Y /Z → Ω1X/Z is
an isomorphism. Then f is étale.
We also have:
Proposition 3.4. a) If f : X → Y is smooth then Ω1X/Y is a locally free
sheaf of finite rank on X.
f
b) If X −
→ Y is étale, then Ω1X/Y = 0.
g
c) If X →
− Y is smooth and i : Z ,→ X is a closed embedding, then the
composite f = i ◦ g : Z → Y is smooth if and only if the sequence
0 → i∗ (IZ /IZ2 ) → i∗ (Ω1X/Y ) → Ω1Z/Y → 0
is exact and locally split.
Proof. a) We may assume that X = Spec B and Y = Spec A. We want to
show that Ω1B/A is projective. We define an operation, “bracketing”: If M
is a B-module let B[M ] be the B-algebra B ⊗ M where (b, m) · (b0 , m0 ) =
(bb0 , bm0 + b0 m). This gives an evident surjection πM : B[M ] → B with
square zero kernel.
Then it is an easy exercise to check that we have a bijection
HomModB (ΩB/A , M ) ' HomA−alg (B, B[M ]).
24
So we have found an adjunction, and it is easy to check that it is functorial. Since showing that ΩB/A is projective is, by definition, equivalent
to showing that there exists a dotted map below for any B-module M 0 :
ΩB/A
?
M
/ M0
|
/0
But by the adjunction, this is equivalent to filling in the diagram below:
B
?
z
B[M ]
/ B[M 0 ]
/0
But since X is smooth, the ?-map exists by definition, so Ω1B/A is projective.
b) If f is étale, the set of liftings is trivial, so by Proposition 3.2 HomR (Ω1X/Y , M ) =
0, hence Ω1X/Y = 0.
c) This was not proven.
Part c) of the proposition gives a “geometric meaning of smooth/étale
maps”. We will now try to explain what this means.
Let f : Spec B → Spec A be smooth. Fix x ∈ Spec B. Assume Ω1B/A is
free of rank r. Choose a surjection
A[x1 , . . . , xn ] B
with kernel I. We than have a split exact sequence of B-modules:
d¯
0 → I/I 2 −
→ Ω1A[x1 ,...,xn ]/A ⊗A B → Ω1B/A → 0.
This implies that there exist n − r functions f1 , . . . , fn−r ∈ I that map
¯ j ) = Pn ∂fj dxi . Because the
to a basis of I/I 2 . Note that we have d(f
i=1 ∂xi
classes of fj form a basis for I/I 2 and the differentials {dxi }ni=1 form a basis
for ΩA[x1 ,...,xn ]/A , the condition that d¯ is a split injection is equivalent to the
∂f
(n − r) × (n − r) minors of the matrix ( ∂xji ) ∈ Mn×(n−r) (B) generating the
unit ideal of B.
25
This tells us that there exist g ∈ A[x1 , . . . , xn ] such that the (n − r) ×
(n − r) minors generate the unit ideal in A[x1 , . . . , xn ][1/g] and such that
the ideal in A[x1 , . . . , xn ][1/g] generated by I is hf1 , . . . , fn−r i.
This proves:
f
Proposition 3.5. Let X −
→ Y be of locally finite presentation. Then f is
smooth if and only if for every x ∈ X there exists an affine neighbourhood
Spec B ⊆ X containing x and an affine neighbourhood f (x) ∈ Spec A ⊆ Y
with f (Spec B) ⊆ Spec A and such that
B ' A[x1 , . . . , xn ]/hf1 , . . . fs i[1/g]
for f1 , . . . , fs , g ∈ A[x1 , . . . , xn ] with s ≤ n and such that the s × s-minors of
the Jacobian generate the unit ideal of B.
If in addition f is étale, then the above holds with s = n.
So a smooth map has at least some similarity with the concept in differential geometry with the same name.
Corollary 3.6. Let A be a ring and f ∈ A[x] a monic polynomial. Then
the finite ring map A → A[x]/(f ) is étale if and only if f 0 maps to a unit in
A[x]/(f ).
Proof. The Jacobian in this case is just the 1 × 1-matrix (f 0 ), and the minors
is just the single element f 0 , and the proposition says that A → A[x]/(f ) is
étale if and only if f 0 generate the unit ideal in A[x]/(f ). But since (f 0 ) is
principal, this happens if and only if f 0 is a unit in the quotient ring.
Example 3.7. Let K be a field and F ∈ K[x] an irreducible polynomial.
Then L := K[x]/(F ) is a finite field extension of K. Then the corollary says
that L/K is étale if and only if F 0 6= 0.
F
x7→xn
Example 3.8. Let A1k −−−−→ A1k be the map sending a point x to its n’th
power. Then the map is étale at x 6= 0 if and only if (n, char(k)) = 1. It is
not étale at any point if n = char(k).
F
Here’s an important corollary. It says that smooth morphisms always
have sections locally in the étale topology.
Corollary 3.9. Let f : X → Y be smooth and let x ∈ X. Then there exists
an étale map π : Y 0 → Y with image containing f (x) and a map s : Y 0 → X
such that f ◦ s = π.
>X
s
Y0
π
26
/Y
f
Proof sketch. After shrinking X, we may assume there exists an étale map
α : X → AnY for some n. The reference is SGA. Let Y 0 be the fiber product
of the diagram
X
0
Y
α
/ An
Y
s
Then Y 0 fits into a diagram Y ← Y 0 →
− X as desired.
√
Remark. So for example, locally n x exists: Just consider the smooth map
x 7→ xn . This will be useful in a later example.
3.2
Invariance of étale sites under infinetesimal thickenings
Let S0 ,→ S be a closed embedding, defined by a nilpotent ideal. Let Ét(S)
be the category having objects étale S-schemes and morphisms S-maps.
Theorem 3.10. The functor
F : Ét(S) → Ét(S0 )
(Z → S) 7→ (Z ×S S0 → S0 )
is an equivalence of categories.
Proof. This is Theorem 3.23 [Mil80].
We skip the proof for now, but philosophically it says that the étale
topology on S is equivalent to the étale topology on S0 , hence when dealing
with sheaves in the étale topology, we can always assume that S is reduced.
27
4
Functors, Hilbert Polynomial, Grassmannians
We begin by recalling something about representable functors.
Recall that a functor F : C op → Set is representable if F ≈ hX for some
X ∈ C.
Definition 4.1. Let F, G be functors C op → Set. We say that a morphism
f
F −
→ G is relatively representable if for all X ∈ C and g : hX → G, the fiber
product
hX ×G F : C op → Set
is representable.
Often we will just say that a morphism is representable. In the future,
we will often not distinguish between
Now let C be the category Aff S of affine schemes over an affine scheme
S. Any affine S-scheme X does of course define a functor hX as usual.
f
Definition 4.2. A morphism of functors F −
→ G is an affine open (closed)
embedding if the following holds:
1. f is relatively representable.
2. For all affine schemes X and natural transformations hX → G, the
map F ×G hX → hX is an open (closed) embedding.
Note that the second condition makes sense only because of the first
condition.
Definition 4.3. A (big) Zariski-sheaf on Aff S is a functor
F : Aff op
S → Set
such that for all U ∈ Aff S , and an open cover {Ui } of U of affine schemes,
the sequence
Q
/Q
/
F (U )
/ i,j∈I F (Ui ∩ Uj )
i∈I F (Ui )
is an equalizer.
Definition 4.4. A morphism of (big) Zariski sheaves g : F → G is surjective
if for every affine scheme U and u ∈ G(U ), there exists an open cover of U ,
{Ui }, such that u|Ui ∈ G(Ui ) is in the image of F (Ui ) for all i ∈ I.
28
Here’s a criterion to check when a functor is represantable in schemes.
Proposition 4.5. A functor F : Aff S → Set is represented by a separated
S-scheme if and only if the following three conditions are satisifed:
1. F is a big Zariski sheaf.
2. The diagonal map ∆ : F → F ×S F is relatively representable and is a
closed embedding.
3. There exists affine schemes {Xi } and morphisms πi : hXi → `
F that are
relatively representable and affine open embeddings such that i∈I hXi →
F is a surjection of Zariski sheaves.
In this case we have an equivalence of categories induced by the Yoneda embedding h_ . On the left side: the category of separated S-schemes. On the
right side: the category of functors F : Aff S → Set that satisfies 1)-3).
Proof. Suppose that X is a separated S-scheme, and let F = hX . We want
to show that 1)-3) is satisified.
1. Let U = ∪Ui where U, Ui are affine. Need to show that a map of
schemes F : U → X is equivalent to a collection of maps {fi : Ui → X}
such that fi|Ui ∩Uj = fj|Ui ∩Uj . But this is obvious.
2. The Yoneda map commutes with products, that is, hX×S X ≈ hX ×S
hX . Thus we can identify the diagonal map ∆ : hX → hX ×S hX with
∆ : hX → hX×S X , which is induced by ∆ : X → X ×S X, which is a
closed embedding since X/S is separable.
3. Let X = ∪Xi be an affine open cover. We get maps hXi → hX , induced
by the inclusions. If T is an affine S-scheme and hT → hX corresponds
to a map of schemes T → X, then the fiber product hT ×hX hXi
is represented by f −1 (Xi ), because the Yoneda embedding commutes
with the fiber product.
Since X/S is separable and S is affine, it follows that each f −1 (Xi )
is an open affine subset of T , thus
` the πi constitutes an open affine
embedding. Then it is clear that i∈I hXi → F is surjective.
We now try to prove the other direction.
29
So suppose F : Aff S → Set is a functor satisfying 1-3). Choose πi : hXi →
F as in 3). We get a diagram
hXi
hXj
πj
πi
/F
Because of the assumptions, the fiber product is represented by an affine
scheme Vij , so get open embeddings Xj ←- Vij ,→ Xi . There is an obvious
isomorphism of functors hXj ×F hXi ' hXi ×F hXj obtained by switching
factors. This induces an isomorphism ϕ : Vij → Vji satisfying the glueing
conditions of schemes. So we can glue to get a scheme X. Then F ' hX .
[[<- The last statement was an exercise...]]
For the equivalence in the proposition, one must show that h_ is fully
faithful, which is part of the statement of the Yoneda lemma.
4.1
The Hilbert polynomial
Let k be a field and j : X ,→ Prk a closed embedding. Let O X (1) be the
inverse image of the Serre twist O Prk (1) under j. Let F be a coherent sheaf
on X.
Definition 4.6. The Euler characteristic of F is the integer
χ(X, F) =
∞
X
(−1)i dimk H i (X, F).
i=0
By a theorem of Grothendieck (see [Har77, Chapter III, Theorem 2.7]),
the higher cohomomology of F vanishes, so the sum is finite. In fact,
H i (X, F) = 0 for i > dim X. We often abbreviate χ(X, F) by χ(F).
By Ex 5.1 in [Har77], we have the following:
Proposition 4.7. Let F, G, H be coherent sheaves on X, fitting into a short
exact sequence:
0 → F → G → H → 0.
Then χ(G) = χ(F ) + χ(H).
Proof. A sort exact sequence of sheaves induces a long exact sequence on
homology. After sorting the terms, we get the result.
30
Corollary 4.8. Let
d
dn−1
d
1
2
0 → F1 −→
F2 −→
· · · −−−→ Fn → 0
P
be an exact sequence. Then ni=1 (−1)i χ(Fi ) = 0.
Proof. Split the long exact sequence into many short exact sequences:
80
0
$
im di+1
di+1
...
di
/ Fi
:
/ Fi+1
;
di
di+1
%
/ Fi+2 · · ·
/
"
im
< di
#
0
0
Then we have
X
X
(−1)i χ(F i ) =
(−1)i (χ(im di ) + χ(im di+1 ))
X
X
=
(−1)i χ(im di ) +
(−1)i−1 χ(im di ) = 0
Hence the corollary is proved.
The next proposition is Exercise 5.2 in Chapter III in [Har77].
Proposition 4.9. There exists a polynomial P (z) ∈ Q[z] such that
χ(F(n)) = P (n)
for all n ∈ Z. In addition, the degree of P is bounded above by the dimension
of X.
Proof. We start by simplifying the situation. We can substitute F with j∗ F
and X with Prk because H i (X, F) = H i (Prk , j∗ F). So we can assume F is a
coherent sheaf on projective space.
By flat base change we can assume that k is algebraically closed.
31
We proceed by induction on the dimension of the support of F. Recall
that supp(F) = {x ∈ X | Fx 6= 0}.
If supp(F) = ∅, then F = 0, so P (n) = 0 works. If however F =
6 0, we
have an exact sequence
·x
r
0 → G → F(−1) −−→
F → H → 0.
If xr is chosen so that the hyperplane xr = 0 does not contain any components of supp(F), it follows that supp(G) and supp(H) have dimension
strictly lower than dim supp(F). Hence by the induction hypothesis, the
difference
χ(Prk , F(n)) − χ(Prk , F(n − 1))
is a polynomial of degree ≤ dim supp(F). The rest follows like in Section 7,
Chapter I, in Hartshorne.
The polynomial P (n) is called the Hilbert polynomial of the O X -module
F. If F = O X , then we say that P (n) is the Hilbert polynomial of X.
By Serre’s vanishing theorem (III, Thm 5.2 in [Har77]), there exists a
natural number n0 ∈ N such that for all i > 0, H i (X, F(n)) = 0 for n > n0 .
So P (n) = dimk H 0 (X, F(n)) for large n. This is the “old” definition of the
Hilbert polynomial.
Exercise 4. Verify that the Hilbert polynomial of Prk is P (n) = n+r
r . Also,
the Hilbert polynomial of Spec(k[t]/(t2 )) is P (n) = 2. Note that this is also
the Hilbert polynomial of two distinct points.
♠
Solution 4. For the first part, note that dimk H 0 (X, O Prk (n)) is just the
number of monomials of degree n. A counting argument shows that there
are n+r
of these.
r
Since X = Spec(k[t]/(t2 )) is affine, all higher cohomology vanish. Hence
P (n) = dimk H 0 (X, O X ), which is just the k-dimension of k[t]/(t2 ) ' k ⊕ tk,
so P (n) = 2.
♥
The next theorem says that in a flat family, the Hilbert polynomial is
constant. In fact, the converse is also true.
Theorem 4.10. Let T be an integral noetherian scheme. Let X ⊆ PrT be a
closed subscheme and let F be a coherent sheaf on X. For each t ∈ T , let
Pt ∈ Q[t] be the Hilbert polynomial of Ft .
f
Then F is flat relative to X −
→ T if and only if Pt is independent of t
(i.e. all fibers have the same Hilbert polynomial).
32
Here’s a diagram of the situation:
/X
Xt
Prk(t)
y
f
%
Spec(k(t))
PrT

/T
Proof. This is proved in Hartshorne, Chapter II, Section 9, as Theorem 9.9.
The theorem says, philosophically at least, that the Hilbert scheme,
parametrizing flat families with Hilbert polynomial P (t), is universal among
families parametrizing flat families.
4.2
The Grassmannian
For more about the Grassmannian (more than will be needed), take a look
at the article [Kle69].
Let k be a field. We will define the Grassmannian G(d, n) for n ≥ d as a
projective k-scheme. We have the the wedge map
(nd)
f : Adn
=
Spec(k[x
])
→
Spec([x
])
=
A
ij i≤d,j≤n
J
k
k
(n)
The coordinates on Akd are {xJ }, where J is an ordered d-tuple, and the
coordinates on Adn
k are xij , elements of a generic d × n-matrix.
The map f is defined dually by f # (xJ ) = det(xij ), which is to be interpreted as taking the determinant of the d × d submatrix of (xJij ) involving
only the columns J.
This gives us a rational map
(n)−1
Pk d
φ : Adn
k
( n)
(n)−1
via composition of f with the standard rational map from Akd to Pkd .
The locus of indeterminacy is V (det(xij )), the zero set of all subdetermi(nd)−1
nants. So we have a well-defined map ϕ : Adn
\V
(det(x
))
→
P
.
ij
k
k
If we set xI = (−1)σ xJ if they differ by a permutation, we see that each
( d )−1
xI gives a well-defined element of H 0 (Pkn , O(1)).
33
The Plücker quadric associated to I, J (both m-tuples) is the zero set of
the following set of equations, for 1 ≤ s ≤ n:
QI,J,s := xI xJ −
d
X
xi1 ...is−1 jl is+1 ···id xj1 ···jl−1 is jl+1 ···jd .
l=1
Then we define the Grassmannian G(d, n) to be the zero set of all the
(n)−1
Plücker quadrics inside Pkd .
Example 4.11. Set d = 2 and n = 4. Then there is only one Plücker
quadric, and it is given by
x12 x34 − x13 x24 + x14 x23
inside P5 . So G(2, 4) is a quadric hypersurface of dimension 4 in P5 .
F
Here comes the geometrical meaning of the Grassmannian:
Proposition 4.12. Denote by U the open set Adn
k \V (det(xij )). Let ϕ be the
n
−1
( )
as above.
map ϕ : U → Pkd
1. ϕ factorizes through G(d, n). That is, there is a commutative diagram
/ G(d, n)
_
U
ϕ
!
(nd)−1
Pk
2. The map ϕ : U → G(d, n) is a locally trivial fiber bundle with fiber
GLd (k), and it gives a bijection between d-dimensional vector subspaces
of k n and k-rational points of G(d, n).
(n)−1
3. Let J be a sorted d-tuple and let UxJ := Pkd \V (xJ ). Then G(d, n) ∩
d(n−d)
UxJ ' Ak
. It follows that G(d, n) is a smooth, projective variety
of dimension d(n − d).
Proof. The first conditions say that the Plücker quadrics define the image of
ϕ. This is a standard, see for example the references in my master’s thesis
[Mey13].
34
For number 2), observe that ϕ is equivariant with respect to left multiplication from GLk (m), the group of invertible d × d-matrices. That is, there
is a commutative diagram:
GLk (d) × U
µ
/U
ϕ
n
d
pr2
U
ϕ
/ P( )−1
k
This almost proves 2). The rest was an exercise.
Thus the Grassmannian is a parameter space, parametrizing d-dimensional
subspaces of the affine space k n .
4.3
The Grassmannian as a functor
Let T be any scheme. We consider the set of all exact sequences
0 → K → O nT → Q → 0
(2)
where K and Q are locally free sheaves on T of rank d and n−d, respectively.
We say that two such sequences are equivalent ∼ if there is a commutative
diagram:
/K
/ On
/Q
/0
0
T
0
'
/ K0
=
/ On
T
'
/ Q0
/0
Definition 4.13. We define a contravariant functor Fd,n : Schop
k → Set. On
objects:
T 7→ {0 → K → O nT → Q → 0}/ ∼,
where the sequence is as above. On maps, just define via pullback.
Theorem 4.14. The Grassmannian G(d, n) represents the functor Fd,n .
Proof. This is proven in the first chapter of János Kollár’s book, “Rational
Curves on Algebraic Varieties”, which were handed out in the lecture.
We can of course generalize this. We can substitute k with an arbitrary
commutative ring A with unity. We can even generalize to the Grassmannian
over any scheme S by glueing on affines. We get GS (d, n). We have a
35
“twisted” version of the functor: Let Fm,E (T ) consist of equivalence classes
of sequences
0 → K → ET → Q → 0
of locally free sheaves on T , where the rank of K is d and ET is the pullback
E via T → S (so E is a sheaf on S).
4.4
Quot and Hilbert schemes
Let E be a coherent sheaf on S, where S is a noetherian scheme. Let X
be a projective S-scheme and let P (z) be a polynomial. Consider an exact
sequence ET → Q → 0. We say that two such sequences are equivalent if
there is a commutatie diagram:
ET
=
ET
/Q
/0
/ Q0
/0
Then we define a functor FE,P (z) :
(
equivalence classes of sequences as above, where Q → T
FE,P (z) (T ) =
is flat, and each Qt have Hilbert polynomial P (z).
On maps, we define FE,P (z) (f ) = f˜, where f˜ comes from the commutative
diagram
X ×S U
U
f˜
f
/ X ×S T
/T
Example 4.15. Exercise as well: Let X = S, P (z) = n − m and let E
be locally free of rank n. Then FE,P (z) = Fm,E = hG(m,n) . This follows from
the fact that Q flat over T = X ×S T with Hilbert polynomial n − m is
equivalent to Q being locally free of rank n − m.
F
Theorem 4.16 (Grothendieck). The functor FE,P (z) is represented by a projective scheme, called Quot(E, P (z)).
Proof. See for example the nice article [Nit05].
36
So what is the Hilbert scheme? The Hilbert scheme is the scheme we
get if we let X = Pnk and E = O X . Since sub-O X -modules are in 1 − 1 correspondence with closed subschemes of X, we see that the Hilbert scheme
parametrizes closed subschemes of Pnk with Hilbert polynomial P (z). So the
theorem says that there is a projective scheme parametrizing closed subschemes of Pnk . This is quite remarkable, as this says that in principle, we
could write down a finite set of equations, such that the solution set correspond to all subschemes with a given Hilbert polynomial.
37
5
Topologies, sites, sheaves
In this section we introduce the notion of a topology on a category. This is
motivated by the insufficiency of the Zariski topology.
5.1
Grothendieck topologies and sites
We must generalize the notion of topology. The reason is that we need more
“open sets” to do interesting things with schemes. Instead of considering
open sets, we consider “coverings”, which are certain classes of maps.
Definition 5.1. Let C be a category. A Grothendieck topology on C consists
of a set Cov(X) of sets of morphisms {Xi → X}i∈I for each X in C, satisfying
the following axioms:
≈
1. If V −
→ X is an isomorphism, then {V → X} ∈ Cov(X).
2. If {Xi → X}i∈I ∈ Cov(X) and Y → X is a morphism in C, then the
fiber products Xi ×X Y exists and {Xi ×X Y → Y }i∈I ∈ Cov(Y ).
3. If {Xi ∈ X}i∈I ∈ Cov(X), and for each i ∈ I, {Vij → Xi }j∈J ∈
Cov(Xi ), then
{Vij → Xi → X}i∈I,j∈J ∈ Cov(X).
Remark. Grothendieck/SGA calls this a pretopology.
A site is just a category equipped with a Grothendieck-topology. So in
a sense, a site is just a very general “space”.
Example 5.2. Let X be a topological space and let Op(X) be the category
of open sets of X and inclusions. If we let Cov(U ) consist of all Ui → U
such that ∪Ui = U , then Op(X) is a site, called the small classical site of
X ∈ Top.
The axioms are fullfilled, because in this case, if U1 , U2 are two open sets,
the fiber product U1 ×U U2 is just the intersection U1 ∩ U2 .
F
Example 5.3 (The global classical site). Now let C be the category Top of
all topological spaces. If U is a topological space, then a covering of U will
be a jointly surjective collection of open embeddings {Ui → U }. Here “open
embedding” must mean “open continuous injective map”, or else condition 1)
is not satisfied.
F
38
Example 5.4 (Big Zariski site). Let S be a scheme and SchS the category
of S-schemes. For (U → S) ∈ SchS , let Cov(U ) be the set {Ui → U } of
maps Ui → U that are open embeddings and such that U = ∪Ui . Since open
immersions are stable under base change, this is a site.
It is called the big Zariski-site of S. Again, open embedding must be
interpreted as a morphism that gives an isomorphism between an open subscheme of U .
F
Example 5.5 (Small étale site). Let Ét(S) be the full subcategory of SchS
consisting of étale maps (U → S). We say that a collection of morphisms
{Ui → U }i∈I is in Cov(U ) if
a
Ui → U
i∈I
is surjective. It is clear from earlier results that this defines a Grothendiecktopology. We call this the small étale site on S.
F
Example 5.6 (Big étale site). Now consider the category SchS . In the
small case, we worked only in the subcategory Ét(S), but now we allow all
morphism (U → S). However, the coverings Cov(U ) = {Ui → U } are now
étale maps.
F
Example 5.7 (The fppf site). On SchS , let {Ui → U }i∈I ∈ Cov(U
` ) if each
Ui → U is flat and locally of finite presentation and such that Ui → U is
surjective. We call this site the fppf site of S (from french fidèlement plat de
présentation finie, “faithfully flat and of finite presentation”).
F
Example 5.8. Let Lis−ét(X) be the full subcategory of SchS with objects
smooth
` maps U → X, and let {Ui → U } ∈ Cov(U ) if every Ui → U is étale
and Ui → U is surjective. This is the lisse-étale site 2 .
F
Example 5.9. We also have the lisse site, which is like the previous example,
except that we demand that each Ui → U to be smooth.
F
Note that Zariski-coverings are étale coverings, and étale coverings are
fppf coverings. So we have natural inclusions
big Zariski site ⊆ big étale site ⊆ fppf site.
The two most important topologies for us will be the étale topology and
the fppf topology.
2
“Lisse” is french for “smooth”.
39
5.2
Presheaves and sheaves
Let F : C op → Set be a presheaf. Let Cˆ denote the category of presheaves on
C.
Definition 5.10 (Sheaf). Let C be a site and F ∈ Cˆ be presheaf.
1. We say that F is separated if for all U ∈ C, and coverings {Ui → U } ∈
Cov(U ), the map
Y
F (U ) →
F (Ui )
is injective.
2. We say that F is a sheaf if the diagram
Q
/
/
F (U )
F (Ui )
/ F (Ui ×U Uj )
is an equalizer for all coverings {Ui → U } ∈ Cov(U ).
If we have a presheaf, we can sheafify to get a sheaf, just as we did as
kids.
Theorem 5.11. The inclusion (sheaves on C) ,→ Cˆ has a left adjoint.
Proof. The proof proceeds in two steps. First one shows that the inclusion
(separated presheaves on C) ⊆ (presheaves on C) has a left adjoint, and
secondly one shows that the inclusion (sheaves on C) ⊆ (separated presheaves
on C) has a left adjoint.
So let F ∈ Cˆ be a presheaf. We want to construct a separated presheaf.
This is the easy step. We just let F s be the presheaf that for each U ∈ C
associates the set F (U )/ ∼ where a, b ∈ F (U ) are equivalent if there exists
a cover {Ui → U } ∈ Cov(U ) such that a, b have the same image under the
map
Y
F (U ) →
F (Ui ).
Now if V → U is a morphism in C, then, by definition, {Ui ×U V → V } ∈
Cov(V ). We have a commutative diagram:
/
F (U )
F (V )
/
Q
Q
F (Ui )
F (Ui ×U V )
40
Thus we see that the composition
F (U ) → F (V ) → F (V )/ ∼
factorizes through F (U )/ ∼, and so F s is indeed a functor, which is obviously
a separated presheaf.
Now let F be a separated presheaf. Let F a be the presheaf that to
each U ∈ C assoicates the set of pairs
Q ({Ui → U
Q}i∈I , {ai }i∈I )/ ∼ where
{Ui → U } ∈ Cov(U ) and {ai } ∈ ker( F (Ui ) ⇒ i,j∈I F (Ui ×U Uj )). We
say that
({Ui → U }i∈I , {ai }) ∼ ({Vj → U }j∈J , {tj })
if ai and bj have the same image in F (Ui ×U Vj ) for all (i, j) ∈ I × J.
It is an exercise in patience to check that this gives us a sheaf.
Now, in good Yoneda-Grothendieck spirit, one realizes that the basic
object of study is not the space itself, but sheaves on it.
Definition 5.12 (Topoi). A topos is a category equivalent to the category
of sheaves on a site 3 .
Definition 5.13 (Morphisms of topoi). A morphism of topoi f : T → T 0
is an isomorphism class of triples (f∗ , f ∗ , φ) where f∗ : T → T 0 and f ∗ :
∼
T 0 → T are covariant/contravariant functors and φ : HomT (f ∗ −, −) −
→
∗
HomT 0 (−, f∗ −) is an isomorphism of bifunctors, that is, f∗ and f are adjoint functors. In addition, we demand that f ∗ must commute with finite
projective limits (in particular, f ∗ preserve fiber products).
There is also the notion of a continuous map of topoi, but we won’t
mention that here.
f
Lemma 5.14. A morphism X −
→ Y in a site C induces a morphism of topoi
g
g
C/X → C/Y .
Proof. First of all, a morphism X → Y induces a morphism C/Y to C/X
given by sending an object (Z → Y ) to the fiber product (Z ×Y X → X).
Given a sheaf E on X, we define f∗ E(Z → Y ) := E(Z ×Y X → X).
Similarly, for a sheaf F on Y , we define f ∗ as f ∗ (Z → X) = F (Z → X → Y ).
One can check that is functorial.
It remains to check that f ∗ and f∗ are adjoint to each other.
3
Plural “topoi”
41
We will denote the different toposes by Xét (sheaves on the small étale
site), XET (sheaves on the big étale site), XZar (sheaves on the small Zariski
site), XZAR (sheaves on the big Zariski site), Xf ppf (sheaves on the big fppf
site of X).
The great thing is that different sites can give rise to equivalent topoi.
Example 5.15. The category of sheaves on Sch is equivalent to the category
of sheaves on AffSch, the category of affine schemes, since sheaves can be
defined locally on affine opens.
F
The main theorem of this section is the following:
Theorem 5.16. Let X → Y be a morphism of schemes. Then hX is a sheaf
in the fppf-topology on SchY .
The theorem says that morphisms of schemes can be glued, even in
topologies much finer than the Zariski topology. In particular, all representable functors are sheaves.
This is a highly non-trivial theorem, and the proof will need several
lemmas. First, we consider the affine case. So let f : A → B be a ring
map and M an A-module. Then MB := M ⊗A B is a B-module. Similarly,
MB⊗A B = M ⊗A (B ⊗A B). By abuse of notation, write f : M → MB for
the map induced by f . Let p1 , p2 : MB → MB⊗A B be the maps induced by
b 7→ b ⊗ 1 and b 7→ 1 ⊗ b, respectively.
Lemma 5.17. If f : A → B is faithfully flat then the sequence
0
/M
f
/ MB
p2
p1
/
/ MB⊗A B
is exact.
Proof. Since B is faithfully flat over A, the lemma is equivalent to the sequence
0 → MB
f0
/ MB⊗ B
A
p02
p01
/
/ M(B⊗A B)⊗A B
being exact. We first show that f 0 is injective. Multiplication B ⊗A B → B
induces a map η : MB⊗A B → MB . Then η ◦ f 0 = idMB , which implies that
f 0 is injective.
Now assume α ∈ MB⊗A B satisfies p01 (α) = p02 (α). Define τ : MB⊗A B⊗A B →
MB⊗A B by multiplying the last two factors. Then τ ◦ p01 = id and τ ◦ p02 =
f 0 ◦ η. Then
α = τ (p01 (α)) = τ (p02 (α)) = f 0 (η(α)).
42
So α is in the image of f 0 , that is, the sequence is exact.
We get as a corollary:
Corollary 5.18. Let X be an affine scheme and assume V → U is a faithfully flat map between affine schemes. Then the sequence
hX (U ) → hX (V ) ⇒ hX (V ×U V )
is exact.
Proof. Let X = Spec R and V = Spec B and U = Spec A. Then the statement is equivalent to the exactness of the sequence
HomA−alg (R, A) → HomA−alg (R, B) ⇒ HomA−alg (R, B ⊗A B).
Let M be the A-module Hom(R, A). Then Hom(R, B) = M ⊗A B and
Hom(R, B ×A B) = MB⊗A B , so the exactness of the above sequence is equivalent to the statement of Lemma 5.17.
We need another lemma. It says that when checking if a functor is a fppf
sheaf, it is enough to consider the case of a single morphism.
Lemma 5.19. Let F : Schop → Set be a Zariski sheaf. Then F is a fppf
sheaf if and only if for any faithfully flat map of locally finite presentation
V → U , the sequence
F (U ) → F (V ) ⇒ F (V ×U V )
is exact.
Proof. Suppose that {Ui → U } ∈ Cov(U ) in the fppf topology. Let V =
`
Ui . Then V → U is a fppf map as in the lemma. We have a commutative
diagram:
/
/ F (V )
F (U )
/ F (V ×U V )
F (U )
/
Q ≈
F (Ui )
/
/Q
i,j
≈
F (Ui ×U Uj )
The exactness of the top row is equivalent to the exactness of the bottom
row.
Here’s another lemma. It says that when checking if F is an fppf sheaf,
it is enough to consider single affine morphisms.
43
Lemma 5.20. Let F be a functor Schop → Set. Then if the two conditions
1. F is a sheaf in the Zariski topology and
2. If V → U is an fppf map of affine schemes, then the sequence
F (U ) → F (V ) ⇒ F (V ×U V )
is exact.
are satisfied, F is a sheaf in the fppf topology.
Proof. Let f : V → U be fppf covering consisting of a single morphism f .
By Corollary 2.5, we can choose a cover {Vi } of V such that each f (Vi ) is
affine and each Vi is quasi-compact. Write each Vi as a finite union ∪a Via of
affine schemes4 . Then we have a diagram:
Q
Q
/
i F (Ui )
i,j F (Ui ∩ Uj )
/
/ F (V )
F (U )
/
Q
Q
/ F (V ×U V )
i,a F (Via )
/
/Q
i,a,b F (Via ×U Vib )
i,j,a,b F (Via ∩ Vjb )
We need to show that the top row is exact. The first column is exact since
the Ui cover U in the Zariski topology, and the second column is exact since
the Via cover V in the Zariski topology. The middle row is exact because
for each i, the Via are a finite number of affines covering Ui , and by Lemma
5.19, this is equivalent to the having just a single covering, but we know that
in the case of affines, this row is exact, by Corollary 5.18.
Now a diagram chase shows that the top row is exact.
We need one more lemma before we can prove Theorem 5.16.
Lemma 5.21. Let f1 : X1 → Y and f2 : X2 → Y be morphisms of schemes.
If x1 , x2 are points of X1 , X2 , respectively, satisfying f1 (x1 ) = f2 (x2 ), there
exists a point z in the fibered product X1 ×Y X2 such that pr1 (z) = x1 and
pr2 (z) = x2 .
4
We need a finite union here, because if Ui are affine schemes, then
if the index set is finite.
44
`
Ui is only affine
Proof. For a point x in a scheme X, write k(x) for the quotient field O X,x /mx .
Set y = f1 (x1 ) = f2 (x2 ) ∈ Y . Then k(y) ,→ k(x1 ) and k(y) ,→ k(x2 )
are two field extensions. The tensor product k(x1 ) ⊗k(y) k(x2 ) is not 0
because the tensor product of two non-zero vector spaces is never zero.
Hence k(x1 ) ⊗k(y) k(x2 ) has a maximal ideal m. The quotiend field K :=
(k(x1 )⊗k(y) k(x2 ))/m is an extension of k(y) containing both k(x1 ) and k(x2 ).
Let U be an affine open in Y containing y. The two composites
fi
Spec K → Spec k(xi ) → Xi −
→U
for i = 1, 2 coincide, so we get a morphism Spec K → X1 ×Y X2 . We take z
to be the image of Spec K in X1 ×Y X2 .
Now we are in position to prove Theorem 5.16.
Proof of Theorem 5.16. We have already proven the statement in the case
X is affine. This is the content of Lemma 5.20.
So write X as a union ∪i Xi of open affine schemes. We first show that
hX is separated (cf. Definition 5.10). Let h : V → U be a covering, and take
two morphisms f, g : U → X such that the two composites V → U → X are
equal.
Since V → U is surjective, the inverse images of Xi coincide, so we can
set Ui = f −1 (Xi ) = g −1 (Xi ), and let Vi = h−1 (Ui ). Then the two composites
Vi
f |Ui
/ Ui
f |Ui
/
/ Xi
coincide, and since Xi is affine, we conclude f |Ui = g|Ui by Lemma 5.20.
This holds for all i, hence f = g.
Now suppose that g : V → X is a morphism such that the two composites
pr1
V ×U V
/
pr2
g
/V
/X
?
?
U
coincide. We need to show that g factors through U . From Lemma 5.21 we
get that g factors through U set-theoretically, and by Proposition 2.4, U has
the quotient topology induced by V → U , so the function f : |U | → |X| (of
topological spaces) is continuous.
45
Set Ui = f −1 (Xi ) and Vi = g −1 (Vi ) for all i. Then the composites
pr1
Vi ×U Vi
pr2
/
/ Vi
g|Vi
/ Vi
/ Xi
coincide. Since Xi is affine, it follows from the affine case of the theorem we
are trying to prove that g|Vi factors uniquely through a morphism f : Ui →
Xi . We have
fi |Ui ∩Uj = fj |Ui ∩Uj : Ui ∩ Uj → X
because hX is separated, hence the fi glue to give the desired factorization
V → U → X.
46
6
Fibered categories
Fibered categories are constructs designed to handle all the different (yet
isomorphic) fiber products. We will see later that stacks are a special kind of
fibered category. In particular, it is a category fibered in groupoids satisfying
a descent condition.
6.1
Fibered categories
Let C be any category. For (almost) all examples here, C will just be the
category SchS of S-schemes in some topology (e.g. the Zariski, étale, fppf
topology).
Definition 6.1. A category over C is a couple (F, p) where F is a category
and p : F → C is a functor.
We say that a morphism φ : α → β in F is cartesian if for all γ ∈ F with
a morphism ψ : γ → β and a factorization
/ p(α) p(φ) / p(β)
5
h
p(γ)
p(ψ)
in C, of p(ψ), then there exists an unique morphism λ : γ → α such that
φ ◦ λ = ψ and p(λ) = h. We draw a diagram to illustrate this:
ψ
F:
γ_
∃!λ
C:
p(γ)
h
/α
_
φ
"
/β
_
p(φ)
/ p(α)
/ p(β)
If φ : α → β is cartesian, then α is called a pullback of β along p(φ).
φ
φ0
Exercise 5. If α −
→ β and α0 −→ β are two pullbacks of β along p(φ),
then there exists an unique isomorphism λ : α → α0 with p(λ) = idp(α) and
φ0 ◦ λ = φ.
♠
Solution 5. This follows from the usual trickery with universal properties.
♥
Definition 6.2. For U ∈ C, let F (U ), the fiber of pF over U , be the category
with objects α ∈ F with p(α) = U (actual equality) and morphisms α0 → α
in F such that p(f ) = idU .
47
Definition 6.3 (Fibered category). We define fibered categories in 1), morphisms between them in 2), and natural transformations between morphisms
in 3).
1. A fibered category over C is a category over C such that for all morphisms f : U → V in C and β ∈ F (V ), there exists a cartesian morphism φ : α → β such that pF (φ) = f . (in particular, α ∈ F (U )).
2. A morphism between fibered categories pF : F → C and pG : G → C is
a functor g : F → G such that
(a) pG ◦ g = pF .
(b) g maps cartesian morphisms to cartesian morphisms.
3. Let g, g 0 : F → G be morphisms of fibered categories. Then a basepreserving natural transformation π : g → g 0 is a natural transformation of functors such that for all α ∈ F , the morphism πα : g(α) →
g 0 (α) in G projects to the identity in C (under pG ). (i.e. πα is a
morphism in G(pF (α)))
This gives us a new category, HOMC (F, G) with objects morphisms of
fibered categories and morphisms base-preserving natural transformations.
Remark. The category of categories fibered over C is an example of a 2category (which can be thought of as a category where the Hom-sets are categories themselves).
It will be useful to single out unique pullbacks. This way there will be
no ambiguity when talking about the fiber products.
Definition 6.4 (Cleavage). A cleavage of a fibered category F → C consists
of a class K of cartesian arrows such that for every f : U → V in C and
η ∈ F (V ), there exists a unique morphism to η that maps to f in C.
A cleavage always exists by the axiom of choice.
Example 6.5. Let f : X → Y be a map of schemes. For T → Y , consider X ×Y T . Let C = SchY . Then define the category F to have objects
quadruples (t, P, a, b) where t : T → Y is a map, P is a scheme, a : P → T
is a map and b : P → X is a map. A morphism (t0 , P 0 , a0 , b0 ) → (t, P, a, b)
is a couple of morphisms (α, β) where α : T 0 → T and β : P 0 → P such
48
that a ◦ β = α ◦ a0 and b ◦ β = b0 . The quadruple (t, P, a, b) should also be a
cartesian square:
b
P
/X
a
T
t
f
/Y
Then we have a functor p : F → C that sends a quadruple (t, P, a, b) to
the morphism t : T → Y . Thus, for t : T → Y , the category F (t : T →
Y ) is the category of cartesian squares as above (with a fixed base). Two
objects in F (t : T → Y ) are uniquely isomorphic, and a choice of one object
corresponds to a choice of fiber product T ×Y X.
The axioms of a fibered category corresponds to the existence of fiber
products.
F
Lemma 6.6. Let p : F → C be a fibered category. Then every morphism
ψ : γ → β can be factorized as
φ
λ
γ−
→α−
→β
where φ is cartesian, and λ is a morphism in F (p(γ)).
Proof. Apply p to ψ to get p(ψ) : p(γ) → p(β). Then β ∈ F (p(β)), so by
definition of a fibered category, there exists a cartesian arrow φ : α → β such
that p(φ) = p(ψ). In particular, p(α) = p(γ), so that α ∈ F (p(γ)). Thus we
have a diagram:
ψ
F :
γ_
∃!λ
C:
p(γ)
id
/α
_
p(α)
φ
p(φ)
"
/β
_
/ p(β)
So that ψ = φ ◦ λ where λ projects down to the identity on p(γ).
g
Lemma 6.7. Let F →
− G be a morphism of fibered categories over C such
that for all objects U ∈ C, the functor gU : F (U ) → G(U ) is full and faithful.
Then g is full and faithful as a functor between F and G.
Proof. Let x, y ∈ F be two objects. Then we have a diagram:
g
HomF (x, y)
pF
)
/ HomG (g(x), g(y))
t
HomC (pF (x), pF (y))
49
pG
We need to show that g induces a bijection on the Hom-sets on the top.
By Lemma 6.6 every morphism ψ : g(x) → g(y) in HomG (g(x), g(y))
factors as
ψ
G:
g(x)
λ
/α
_
C:
x
φ
!
/ g(y)
_
/ y
Since g was a morphism of fibered categories, the cartesian morphism φ
comes from a morphism α0 → y, up to unique isomorphism, by Exercise 5.
So we can write α = g(α0 ). Thus by the assumption λ = g(γ) for a unique
γ ∈ HomF (x, y). Thus g is full, and it is faithful because any other φ differs
by a unique isomorphism from HomF (x, x).
g
Definition 6.8. A morphism F →
− G of fibered categories over C is an
h
equivalence if there exists G −
→ F and base-preserving isomorphisms such
that
h ◦ g ' idF
and
g ◦ h ' idG
g
Proposition 6.9. A morphism F →
− G over C is an equivalence if and only
if for all U ∈ C, the functor gU : F (U ) → G(U ) is an equivalence.
Proof. A morphism is an equivalence if it is fully faithful and essentially
surjective. Since morphisms over C are base-preserving, it is enough to chech
on fibers.
Remark. You might say that equivalence can be checked “pointwise” or
“fiberwise”.
6.2
2-Yoneda-lemma
Let pF : F → C be a fibered category. Let X ∈ C. Consider C/X (the slice
category of X in C). We have an evident forgetful functor C/X → C.
We have a map
α : HOMC (C/X, F ) → F (X)
g
(C/X →
− F ) 7→ g(idX )
that is a morphism of fibered categories.
50
Lemma 6.10 (2-Yoneda). The map α is an equivalence of categories.
Proof. We want to define a map in the other direction: Let η : F (X) →
HOMC (C/X, F ) be defined as follows. For x ∈ F (X), let
ηx : C/X → F
be given by (ϕ : Y → X) 7→ ϕ∗ x. Here ϕ ∈ HomC (Y, X) and ϕ∗ x is a choice
of pullback of x ∈ F (X) along pF . This follows from the first property of
fibered categories in Definition 6.3. See the diagram below:
ϕ_∗ x
F :
pF
C:
Y
∃ψ
ϕ
/x
_
pF
/X
Now, given a morphism
Y0
/Y
ϕ0
ϕ
X
in C/X, define ηx () : ϕ0∗ x → ϕ∗ x to be the uniquely defined morphism in
the diagram
/ ϕ∗ x
/( x
ϕ0∗ x
∃!
Thus we have defined ηx . It is a morphism of fibered categories. Compositions of maps need to be checked, though.
f
Note: A morphism x0 −
→ x in F (X) induces a morphism of functors
ηf : ηx0 → ηx : If ϕ : Y → X ∈ HomC (Y, X) and ϕ∗ x0 → ϕ∗ x is a pullback,
let ηf (ϕ) be the unique (since ϕ∗ x is a pullback) morphism ϕ∗ x0 → ϕ∗ x such
that the diagram
ϕ∗ x0
x0
ηf (ϕ)
f
/ ϕ∗ x
/ x
commutes. This gives the desired morphism of functors η(f ). Thus we have
f
defined η on objects (ηx ) and on morphisms (x0 −
→ x). This defines η.
Now consider the composition
η
α
F (X) −
→ HOMC (C/X, F ) −
→ F (X)
51
It sends x ∈ F (X) to id∗X , which is canonically isomorphic to x. This gives
an isomorphism of functors α ◦ η ' idF (X) .
On the other hand, consider η ◦ α. This composition sends a function
f : C/X → F to the functor C/X → F that for (ϕ : Y → X) ∈ HomC (Y, X)
associates ϕ∗ f (idX ). This functor is canonically isomorphic to f . (check
this!)
Thus if F is some moduli problem, the lemma says that the elements of
F (X) (i.e. the solutions of the moduli problem), correspond to maps to the
moduli problem.
So when we have defined stacks, this will imply that a stack is a fine
moduli space for the moduli problem it represents.
Corollary 6.11. Let X, Y ∈ C be objects. Then the functor
HOMC (C/X, C/Y ) → HomC (X, Y )
that maps f to f (idX ) is an equivalence of categories.
Remark. If X ∈ C and F → C is a fibered category, we will often write
X → F to denote an objects in F (X). (“in good Yoneda lemma tradition”)
6.3
Categories fibered in Set
The objects of Set can be identified with categories with no non-trivial structure. That is, one can identify a set X with the category X whose only
morphisms are the identity morphisms.
Definition 6.12. A category fibered in Set is a fibered category p : F → C
such that for all U ∈ C, the category F (U ) is a set.
Lemma 6.13. Let q : G → C be a fibered category and let p : F → C be a
category fibered in Set. Then the category HOMC (G, F ) is a set.
Proof. Let f, g : G → F be morphisms of fibered categories and α : g → f
a morphism in HOMC (G, F ). Then, for all x ∈ G over X ∈ C, we have a
morphism αx : f (x) → g(x) (in F (X)).
Now, if F is fibered in Set, αx must be the identity. But then f (x) = g(x)
for all x ∈ G. But then f = g. So any two objects with a morphism between
them are equal, hence HOMC (G, F ) is a set.
Let F : C op → Set be a presheaf over C. We will construct a fibered
category F over C. The category F has objects pairs (U, x) with U ∈ C and
52
x ∈ F (U ). The morphisms g : (U 0 , x0 ) → (U, x) are given as a morphism
g : U 0 → U such that g ∗ x := F (g)(x) = x0 in F (u0 ). The morphism F → C
is given by (U, X) 7→ U .
It is an easy exercise in definition-hunting to see that this is indeed a
fibered category.
Proposition 6.14. We have an equivalence of categories
Γ : PreSheavesC → {categories fibered in sets over C}
defined by F 7→ F .
Proof. Now given a category p : F → C fibered in Set, define
F : C op → Set
U 7→ F (U )
Thus we have defined a natural transformation Σ (things need to be
checked though) from categories fibered in Set over C to PreSheavesC such
that Σ ◦ Γ = idPreSheavesC .
On the other hand, it is easy to see that Σ is an equivalence in each fiber
(there are no morphisms to check on!), so by Proposition 6.9 it follows that
Σ is an equivalence.
Remark. Schemes give rise to fibered categories because of the embedding
Sch ,→ PreSheavesSch given by X 7→ hX . In particular, schemes give rise to
fibered categories with no automorphisms in the fibers.
So a fibered category p : Groupoids → Sch without automorphisms is the
same as a scheme, in the same way that a scheme without gluing data is the
same as an affine scheme.
53
7
Groupoids and descent
We define groupoids and show that fiber products of groupoids exist. After
that we define what we mean by descent, prove some lemmas, and state the
main theorem.
7.1
Groupoids
Definition 7.1. A groupoid is a category where all morphisms are isomorphisms.
Example 7.2 (The fundamental groupoid). Let X be a topological space.
Then Π1 (X) is the category with objects the points of X, and morphisms the
paths betweens points of X, up to homotopy equivalence. This is a groupoid
because any path has an inverse path. Then the usual fundamental group
π1 (X, x) based at x ∈ X is just AutΠ1 (X) (x).
F
Definition 7.3. A fibered category p : F → C is fibered in groupoids if every
fiber F (U ) is a groupoid.
Remark. In particular, every scheme give rise to a category fibered in groupoids,
because each fiber is just a set, and identity morphisms are isomorphisms.
Example 7.4. Let C be the category Top of topological spaces and let F
be the category consisting of triples (X, x, π1 (X, x)) where X ∈ Top is a
topological space, x ∈ X and π1 (X, x) is the fundamental group based at x.
Define maps between (X, x, π1 (X, x)) and (Y, y, π1 (Y, y)) to consist of a map
X → Y ∈ HomTop (X, Y ) such that f (x) = y. This induces a corresponding
map on the fundamental groups by composition: f∗ : π1 (X, x) → π1 (X, y).
Define p : F → C by (X, x, π1 (X, x)) 7→ X. Then the fiber F (X) over X
is just the fundamental groupoid of X, so we have an example of a fibered
category fibered in groupoids.
F
Lemma 7.5. Let
F
/C
F0
be categories fibered in categories. Assume further that F 0 is fibered in
groupoids. Then the category HOMC (F, F 0 ) is also a groupoid.
54
Proof. Let f, g : F → F 0 be morphisms of fibered categories and let α : f → g
be a morphism in HOMC (F, F 0 ). To show that α is an isomorphism, it is
enough to do it on each fiber, by Proposition 6.9, but αx : f (x) → g(x)
is a morphism in F 0 , hence an isomorphism. So α is an isomorphism, so
HOMC (F, F 0 ) is a groupoid by definition.
Definition 7.6 (The “slice groupoid”). Let p : F → C be fibered in groupoids.
For X ∈ C, we get a new category fibered in groupoids as follows: The notaf
tion is p/X : F/X → C/X. The objects of F/X are pairs (y, p(y) −
→ X) with
y ∈ F and f a morphism in C. The morphisms are morphisms g : y 0 → y in
F such that f ◦ p(g) = f 0 .
f
Then p/X sends (y, p(y) −
→ X) to f : p(y) → X. For a morphism
Y → X, the fiber (F/X)(Y → X) is just F (X). This is a groupoid since F
was fibered in groupoids.
The next definition will be important in the definition of stacks.
Definition 7.7 (The isomorphism presheaf). Let p : F → C be a category
fibered in groupoids. For X ∈ C, let x, x0 ∈ F (X). Then we get a presheaf
Isom(x, x0 ) : C op /X → Set defined as follows. For an object Y ∈ C/X, send
f
For each morphism Y −
→ X, choose pullbacks f ∗ x and f ∗ x0 (elements in
F (Y )) and define
f
Isom(x, x0 )(Y −
→ X) := HomF (Y ) (f ∗ x, f ∗ x0 ) ∈ Set
For a morphism g : Z → Y in C/X, the restriction map is defined by
g∗
HomF (Y ) (f ∗ x, f ∗ x0 ) −→ HomF (Z) (g ∗ f ∗ x, g ∗ f ∗ , y) ' HomF (Z) ((f g)∗ x, (f g)∗ y).
The isomorphism to the right comes from F being a fibered category and
pullbacks being unique up to unique isomorphism.
g
f
Exercise 6. For a composition of Z →
− Y and Y −
→ X we have a natural
map g ∗ : Isom(x, x0 )(Y → X) → Isom(x, x0 )(Z → X).
♠
Solution 6. Hint: By choice of cleavage there are unique isomorphisms
(f g)∗ x ' g ∗ f ∗ x.
♥
Note that when x = x0 we get a presheaf of groups: Autx : (C/X)op →
Grps, and furthermore, the set Isom(x, x0 )(Y → X) is a torsor under the
action of Autx .
55
7.2
Fiber products of categories fibered in groupoids
Suppose we have a diagram of groupoids.
G1
g
G2
f
/G
Then the fiber product G1 ×G G2 of the diagram exists, and it is a groupoid.
Its object are triples (x, y, σ) where x ∈ G1 and y ∈ G2 and σ : f (x) → g(x)
is a morphism in G. The morphism (x0 , y 0 , σ 0 ) → (x, y, σ) are pairs (a, b)
where a : x0 t x is an isomorphism and b : y 0 → y is an isomorphism, such
that the diagram
σ0
f (x0 )
/ g(y 0 )
f (a)
f (x)
g(b)
/ g(y)
σ
commutes. There are evident functors pj : G1 ×G G2 → Gj for j = 1, 2 and a
natural isomorphism of functors Σ : f ◦ p1 → g ◦ p2 . (check this!)
In addition, the triple (G1 ×G G2 , pi , Σ) has a universal property. Suppose
that H is another groupoid and α : H → G1 , β : H → G2 and γ : f ◦ α →
g ◦ β is an isomorphism of functors. Then there exists a (unique) triple
(h : H → G1 ×G G2 , λ1 , λ2 ) such that λ1 : α → p1 ◦ h and λ2 : β → p2 ◦ h are
isomorphisms of functors, and such that the diagram
f ◦α
f (λ1 )
/ f ◦ p1 ◦ h
γ
g◦β
g(λ2 )
Σ◦h
/ g ◦ p2 ◦ h
commutes.
Now let C be a site, and let
F2
F1
d
F3
~
c
be categories fibered in groupoids over C. Suppose further that G is a category fibered in groupoids over C and that we have functors α : G → F1 and
β : G → F2 and an isomorphism γ : c ◦ α → d ◦ β.
56
The triple (α, β, γ) is equivalent with an object in
HOMC (G, F1 ) ×HOMC (G,F3 ) HOMC (G, F2 ).
Such a triple defines in turn a for any category fibered in groupoids H a
morphism of groupoids
∗ : HOMC (H, G) → HOMC (H, F1 ) ×HOMC (H,F3 ) HOMC (H, F2 )
(h : H → G) 7→ (α ◦ h, β ◦ h, γ ◦ h)
Proposition 7.8.
1. There exists a quadruple (G, α, β, γ) such that for
all H fibered in groupoids, the map ∗ above is an isomorphism.
2. If (G0 , α0 , β 0 , γ 0 ) is as above, then there exists a triple consisting of
(F, u, v) where F : G → G0 is an equivalence of fibered categories and
u : α → α0 ◦f and v : β → β 0 ◦f are isomorphisms of fibered categories.
In addition, the diagram
c◦u /
c◦α
γ
d◦v /
d◦β
c ◦ α0 ◦ F
γ0
d ◦ β0 ◦ F
commutes. Further, if (F 0 , u0 , v 0 ) is another such triple, there exists a
unique isomorphism σ : F 0 → F such that
α
u0
u
/ α0 ◦ F 0
"
σ
β
v0
v
α0 ◦ F
/ β0 ◦ F 0
"
σ
β0 ◦ F
We will write F1 ×F3 F2 for the fibered category in part 1 of the proposition. We skip the proof.
7.3
Descent theory
A stack is morally a sheaf of categories fibered in groupoids over C. What do
we mean by a sheaf? The answer lies in descent theory á la Grothendieck.
The situation is that we want some properties to be “local” in the some
“topology”, but for this to make sense, we have to prove that they are local,
which is the point of descent theory.
57
Let C be a category with fiber products and let p : F → C be a fibered
category, and choose a cleavage5 .
We will define a category of descent data. We will say (see definitions
below), that a morphism X → Y is of effective descent if giving the fibered
category on “open sets” in a covering plus some glueing data satisfying a
cocycle condition is equivalent to giving the fibered category on the space
being covered (here Y ).
For a morphism X → Y in C, we define the F (X → Y ): The objects
'
are pairs (E, σ) with E ∈ F (X) and σ : pr∗1 E −
→ pr∗2 E an isomorphism in
F (X ×Y X) such that in F (X ×Y X ×Y X) we have a commutative diagram
pr∗13 pr∗1 E
pr∗12 pr∗1 E
pr∗13 ◦σ
pr∗12 ◦σ
x
&
pr∗13 pr∗2 E
pr∗12 pr∗2 E
pr∗23 pr∗2 E o
pr∗23 ◦σ
pr∗23 pr∗1 E
,which we call the cocycle condition. The equality signs are the canonical
isomorphisms coming from the choice of cleavage. Remember to think of the
fiber fiber products as intersections.
A morphism between pairs (E 0 , σ 0 ) and (E, σ) consists of a morphism
g : E 0 → E in F (X) such that
pr∗1 E 0
pr∗1 ◦g
σ0 '
' σ
pr∗2 E 0
/ pr∗ E
1
pr∗2 ◦g
/ pr∗ E
2
The isomorphism σ is called descent data for E.
Remark. Notice that the cocycle condition is a technical way of writing
(pr∗|23 ◦ σ) ◦ (pr∗12 ◦ σ) = (pr∗13 ◦ σ).
Or better:
σ|23 ◦ σ|12 = σ|13 .
Which is just the usual cocycle condition.
5
we say p is cloven.
58
We have a natural functor
f
: F (Y ) → F (X −
→Y)
(3)
For E0 ∈ F (Y ), choose two pullbacks pr∗1 f ∗ E0 and pr∗2 f ∗ E0 along the projection X ×Y X → Y . Then, because of the chosen cleavage, there is a
unique isomorphism σcan : pr∗1 f ∗ E0 → pr∗2 f ∗ E0 . Then we define on objects
by sending E0 to the pair (f ∗ E0 , σcan ).
It is left to the reader to check that the pair (E, σcan ) actually satisfy the
cocycle condition.
Remark. One can similarly define the descent category F ({Xi → Y }) for
a collection {Xi → Y }i∈I . This is done completely analogously. However,
they are very often equivalent, as the following lemma shows:
Lemma
7.9. Q
Suppose that coproducts exist in C and that the natural functor
`
an equivalence of categories. Let {Xi → Y } be
F ( i Xi ) → i F (Xi ) is `
morphisms in C. Let Q = Xi . Then the map
: F (Y ) → F ({Xi → Y })
is an equivalence if and only if
: F (Y ) → F (Q → Y )
is an equivalence.
Proof. Exercise. (do this later!!)
Definition 7.10. For an object (E, σ) ∈ F (X → Y ), we say that the descent
data σ is effective if (E, σ) is in the essential image of .
Definition 7.11. We say that a morphism X → Y is of effective descent
for F if : F (Y ) → F (X → Y ) is an equivalence of categories.
Algebraic geometers like to prove descent theorems. In one case it is easy,
and this if the map X → Y has a section. So let p : F → C be a fibered
f
category and X −
→ Y a morphism in C.
Proposition 7.12. If there is a section s : Y → X, then f is an effective
descent morphism.6
6
Recall: a section s : Y → X of f : X → Y is a morphism such that f ◦ s = idX
59
Proof. We must show that the functor
: F (Y ) → F (X → Y )
is an equivalence of categories. There are three things to see. That it is
faithful, that it is full, and that it is essentially surjective.
Let η : F (X → Y ) → F (Y ) be defined by sending an object (E, σ) to
the pullback s∗ E.
Showing faithfullness is easy. One sees that the composition η ◦ (E) =
∗
∗
s f E, but this is canonically isomorphic to (f s)∗ E = id∗X E ≈ E. So has
an essential left inverse, hence is faithful.
[[Remains to show fullness and essential surjectivity.]]
60
8
Descent theorems
In this lecture we will prove various descent theorems, which roughly guarantee that many of the objects we work with in the future actually will be
stacks.
8.1
Descent for sheaves on a site
] the category
Let C be a site. For X ∈ C, we have an associated topos C/X,
of sheaves on X.
We define a new fibered category over a site C, and call it Sh. Its objects
]
are pairs (X, E), where X ∈ C and E ∈ C/X.
A morphism (X, E) → (Y, F ) is given by a pair (f, ) where f : X → Y
]
is a morphism in C and : E → f ∗ F is a morphism in C/X.
Notice that if we have a composition
(f,)
(g,ρ)
(X, E) −−−→ (Y, F ) −−−→ (Z, G)
we have that the composition is defined as the pair (g ◦ f ) : X → Z and the
f ∗ ρ ◦ , i.e. by the diagram
f ∗ρ
!
E→
− f ∗ F −−→ f ∗ g ∗ G = (g ◦ f )∗ G.
Let p : Sh → C be defined by (X, E) 7→ X.
Exercise 7. This give us a fibered category with fiber over X ∈ C given by
]
Sh(X) = C/X.
♠
♥
Solution 7. Easy.
Now we are able to state the main theorem of this section. It is also time
to recall the definition of a morphism of topoi, i.e. Definition 5.13.
Theorem 8.1. Any covering in C is an effective descent morphism for Sh.
More precisely, the functor : Sh(Y ) → Sh(X → Y ) is an equivalence of
categories.
Proof. Hopefully to be understood and TeXed some day. The proof was long
and technical.
The theorem has a nice corollary, which – according to the Østvær – is
difficult to prove my other means:
61
Corollary 8.2. Let X, Y be S-schemes. Let S 0 → S be a fppf-cover, and put
S 00 = S 0 ×S S 0 , X 0 = X ×S S 0 and X 00 = X ×S ×S 00 . And also Y 0 = Y ×S S 0
and Y 00 = Y ×S S 00 . We have two projections pr1 , pr2 : S 00 → S 0 .
Suppose f 0 : X 0 → Y 0 is an S 0 -morphism such that pr∗1 f 0 = pr2∗ f 0 : X 00 →
00
Y . Then f 0 is induced by a unique morphism f : X → Y .
Proof. We have fppf-sheaves hX , hY induced by X, Y . Let C be the category
of S-schemes with the fppf-topology.
A morphism f 0 such that pr∗1 f 0 = pr∗2 f 0 is equivalent to a morphism
(hX 0 , σcan ) → (hY 0 , σcan )
in Sh(S 0 → S). But the theorem says that such a morphism is induced by a
unique morphism hX → hY . Now apply the Yoneda lemma.
8.2
Descent for quasi-coherent sheaves
Let S be a scheme and let C be the fppf site of S. That is, the objects of
C are S-maps and the coverings are jointly surjective morphisms {Ui → U }
that are fppf.
Then the sheaf O is a presheaf of rings on C. It assigns for every Sscheme T the set of global sections Γ(T, O T ). This is a sheaf because it is
represented by a scheme A1S (and all schemes are sheaves, by Theorem 5.16).
Now let Qcoh(S) be the category of quasi-coherent sheaves on S. We
will show (among other things) that the category of quasi-coherent sheaves
in the Zariski-topology is equivalent to the category of quasi-coherent sheaves
in the fppf topology.
Let F ∈ Qcoh(S). This gives us a presheaf Fbig of O-modules on C (recall
f
that this is just a functor C op → Set7 ). It sends an S-scheme T −
→ S to the
∗
∗
set Γ(T, f F ) = f F (T ).
Lemma 8.3. The presheaf Fbig is a fppf sheaf.
Proof. By Lemma 5.20 we need to check two things: 1) That Fbig is a Zariski
sheaf. This is clear by childhood knowledge. And 2), that the familiar
equalizer is true for all fppf maps V → U of affine schemes.
So let Spec B → Spec A be fppf. Let M be the A-module corresponding
to pullback of F to Spec A. Then the sheaf condition on Fbig is equivalent
to the exactness of the diagram
0
7
/M
/ M ⊗A B
Well, not really Set, but Mod.
62
/
/ M ⊗A (B ⊗A B).
But this was shown in Lemma 5.17.
Now let G be a sheaf of O-modules on C. Thist just means that for every
scheme T , the set G(T ) is an O(T )-module.
Given a scheme map T → S we get an element GT := G(T ) ∈ Qcoh(T )
by applying T to G. In particular, for F ∈ Qcoh(S) we get an isomorphism
F ' (Fbig )S .
In fact we have an adjunction:
Lemma 8.4. Restriction induces an isomorphism:
'
HomO (Fbig , G) −
→ HomO S (F, GS )
So “biggifying” is left adjoint to “localizing”.8
Proof. Exercise with hint. We can assume that S is affine. Write F as a
cokernel
F2 → F1 → F → 0,
where the Fi ’s are direct sums of copies of O S . Claim: F 7→ Fbig is right-exact
(this is probably because tensoring is right-exact). So we get a commutative
diagram:
0
/ HomO (Fbig , G)
/ HomO (F1,big , G)
/ HomO (F2,big , G)
0
/ HomO (F, GS )
S
/ HomO (F1 , GS )
S
/ HomO (F2 , Gs )
S
The vertical arrows are restriction maps (i.e. applying S to the sheaves
Fbig , G, etc). If the two right arrows are isomorphisms, then the left arrow
is also. So we can assume that F is a direct sum of copies of O S , but if the
statement is true for a direct sum, it is also true if F = O S . So we assume
F = OS.
So we are reduced to proving that there is a bijection
'
HomO (O, G) −
→ HomO S (O S , GS ).
The right hand side is equal to the set Γ(S, GS ) of global sections of GS .
The left hand side is equal to lim Γ(X, GX ). But C has a terminal object,
←−
namely the identity morphism S → S, so the left hand side is in fact also
equal to Γ(S, GS ).
8
Anyone have a better name for the operation ?big ?
63
Definition 8.5. A big quasi-coherent sheaf on S is a sheaf of O-modules F
on C such that:
1. For all S-schemes T ∈ SchS , the sheaf FT on TZar is quasi-coherent.
g
2. For any morphism T 0 →
− T in C, the morphism g ∗ FT → FT 0 is an
isomorphism.
Remark. The map in 2) is the familiar pullback of quasi-coherent map.
This gives us a category Qcoh(Sf ppf ) of big fppf sheaves on S. This is
in contrast to the situation in Hartshorne, where we worked in the category
Qcoh(SZar ) of quasi-coherent sheaves on the small Zariski site of S.
Our discussions above showed that if F ∈ Qcoh(SZar ), then Fbig ∈
Qcoh(Sf ppf ). By definition, we have an equivalence of categories:
Lemma 8.6. We have an equivalence of categories:
Qcoh(SZar ) → Qcoh(Sf ppf )
F 7→ Fbig
GS →7 G
Proof. One direction follows since F = (Fbig )S . The other direction follows
by point 2) in the definition above, namely, that (GS )big ' G.
Let us define a new category QCOH: Its objects are pairs (T, E) where
T is a scheme and E is quasi-coherent sheaf in Qcoh(TZar ). A morphism
between two pairs (T 0 , E 0 ) → (T, E) is a pair (f, ) where f : T 0 → T and
: E 0 → f ∗ E is a morphism between the quasi-coherent sheaves.
We have an evident forgetful functor p : QCOH → Sch.
Exercise 8 (Exercise-Proposition). This gives us a category QCOH fibered
over schemes and the fiber over T ∈ Sch, QCOH(T ) is Qcoh(TZar ).
♠
Solution 8. This mounts down to saying that fiber products of quasicoherent sheaves exist. It does.
♥
f
Theorem 8.7 (Descent for quasi-coherent sheaves). Let X −
→ Y be a fppf
cover of schemes. Then f is an effective descent morphism for QCOH.
64
Proof. If f is quasi-compact and quasi-separated, then f∗ preserves quasif
coherence. In that case, the functor η : Sh(X −
→ Y ) → Sh(Y ) given by
f
(E, σ) 7→ f∗ E will be an inverse for : QCOH(Y ) → QCOH(X −
→ Y ). This
is because fppf implies that f∗ f ∗ is an isomorphism, i.e. Lemma 2.9.
So we want to reduce to that case. So let (E, σ) ∈ QCOH(X → Y ). We
first show that (E, σ) is in the essential image of . Since f is fppf we can
choose a Zariski cover Y = ∪Yi of Y such that f −1 (Yi ) = ∪j Xij where each
Yi is affine and Xij is quasi-compact with f (Xij ) = Yi for all j. Thus for
all i, j we have a quasi-separated and quasi-compact cover Xij → Yi . So
the restriction of (E, σ) to Xij is induced by pullback from a unique object
Fij ∈ QCOH(Yi ).
For indices i, j we have a commutative diagram
QCOH(Yi )
QCOH(Xij
}
t
`
Xij → Yi )
res
res
QCOH(Xij → Yi )
*
!
QCOH(Xij → Yi )
There is a unique isomorphism σijj 0 : Fij → Fij 0 , so Fij is not dependent
upon j, up to isomorphism. So we get an equivalence of categories
(∗)
i : QCOH(Yi ) −−→ QCOH(f −1 (Yi ) → Yi )
Now, an element in QCOH(X → Y ) is, in this cover, equivalent to the
following data: a pair ({(Ei , σi )}, {αii0 }), where (Ei , σi ) ∈ QCOH(f −1 (Yi ) →
Yi ) and for two indices i, i0
'
αii0 : (Ei , σi )|Yi ∩Yi0 −
→ (Ei0 , σi0 )|Yi ∩Yi0
is an isomorphism in QCOH(f −1 (Yi ∩Yi0 ) → Yi ∩Yi0 ) that satisfies the cocycle
condition for triples, but by the equivalence (∗) above, this is equivalent to
the data of a pair ({Ei }, {βii0 }) where Ei is a quasi-coherent sheaf on Yi
and for indices i, i0 , we have an isomorphism in QCOH(Yi ∩ Yi0 ) between
'
βii0 : Ei −
→ Ei0 . But by standard glueing of sheaves (see [Har77]), we get a
sheaf in QCOH(Y ).
65
8.3
Descent for closed subschemes
f
Let X −
→ Y be a fppf cover. Let pri : X ×Y X → X for i = 1, 2 be the two
projections. Let Z ,→ X be a closed subscheme and let pr∗i Z ,→ X ×Y X be
Z ×X,pri (X ×Y X).
Then:
Proposition 8.8. We have a bijection between closed subschemes W ,→ Y
and closed subschemes Z ,→ X such that pr∗1 Z = pr∗2 Z.
The bijection is given by sending W to its inverse image f −1 (W ).
Proof. There is a 1-1 correspondance between closed subschemes and quasicoherent ideal sheaves, [Har77, Chapter II, §5]. So it is enough to show that
the pullback map of quasi-coherent sheaves induces a bijection between the
set of quasi-coherent ideal sheaves ⊆ O Y and quasicoherent ideal sheaves
J ⊆ O X such that pr∗1 J = pr∗2 J . But this follows from descent for quasicoherent sheaves, i.e. Theorem 8.7.
This has the consequence that the fibered category F consisting of pairs
(Y, X) where Y is a closed subscheme of Y and the forgetful morphism is
(Y, X) 7→ X, is a stack. The fibers F (X) are the set of all closed subschemes of X (notice that this is a set, because there are no non-identity
automorphisms of objects of F (X).
8.4
Descent for open embeddings
This turns out to be almost trivial. We don’t need to apply any heavy
theorems.
So let Op be the category with objects pairs (X, U ) where X is a scheme
and U is an open subscheme of X. A morphism in Op is just a scheme map
f
X0 −
→ X such that U 0 ⊆ f −1 (U ).
Exercise 9. The evident map p : Op → Sch gives us a fibered category. ♠
Solution 9. Note first that the fibers Op(X) are just the set of open subschemes of X. Let f : X → Y be a morphism of schemes, and let U be an
open subscheme of Y . Then the ordinary fiber product of schemes gives us
a scheme U ×Y X. But topologically this is just f −1 (U ), which is open (see
Exercise 7.1B in [Vak13]).
So for every map in Sch we get a fiber product, and this is just the
condition in Definition 6.3.
♥
Here’s the statement:
66
f
Proposition 8.9. Any fppf cover S 0 −
→ S is an effective descent morphism
for Op.
Proof. Let S 00 = S 0 ×S S 0 . We must show that if U 0 ⊆ S 0 is an open subset
−1
0
0
00
0
−1 (U ) for a unique open
with pr−1
1 (U ) = pr2 (U ), in S , then U = f
U ⊆ S.
We first observe that uniqueness is clear since f is surjective. This is just
the set-theoretic fact that f surjective implies f (f −1 (U )) = U for all U ⊆ S.
So we prove existence. Since f is fppf, it is an open map, so U := f (U 0 )
is an open subset of S, so at least U 0 ⊆ f −1 (U ). Suppose now f −1 (U )\U 0 .
Then there is some y ∈ U 0 with f (x) = f (y). But then the pair (x, y) ∈ S 00
−1
0
0
will be in pr−1
2 (U ) but not in pr1 (U ). But this contradicts the hypothesis!
So U 0 = f −1 (U ).
Remark. Note that the only property of f : S 0 → S we used was that it was
an open map. So we could have gotten away with assuming, for example,
that f was just flat and locally finitely presented.
8.5
Descent for affine morphisms
f
We have a category Aff: its objects are affine scheme maps X −
→ Y and the
morphisms are commutative diagrams
/X
X0
f0
f
/Y
Y0
There is an obvious functor p : Aff → Sch, given by sending a morphism
(X → Y ) to its target Y .
s
Proposition 8.10. Let S 0 →
− S be a fppf cover. Then s is of effective descent
for Aff.
Sketch proof. This follows from effective descent for modules and noting that
tensor products of modules are preserved (where?).
8.6
Descent for quasiaffine morphisms
f
Recall that a morphism X −
→ Y is quasi-affine if there exists a factorization
X

/W
f
Y
67
g
where the top arrow is an open embedding and g is affine. We can then form
the category QAff of quasi-affine morphisms. The morphisms are commutative squares of such. Again, it is easy to see that we get a fibered category
p : QAff → Sch.
Proposition 8.11. Any fppf cover S 0 → S is of effective descent for QAff.
Sketchy proof. This follows from descent for open embeddings and for affine
morphisms.
8.7
Descent for polarized schemes
f
Let Pol be the category consisting of pairs (X −
→ Y, L) where f is a proper
morphism and L is relatively ample invertible sheaf on X. This means that
for all affine opens V ⊆ Y , the restriction L|f −1 (V ) is ample on X. This
means that for all coherent sheaves F on X, there exists an integer n0 > 0
such that for all n ≥ n0 , the sheaf F ⊗L⊗n is generated by global sections.
For more on this, see [Har77].
f0
f
The morphisms (X 0 −→ Y 0 , L0 ) → (X −
→ Y, L) in Pol are triples (, a, b)
where a, b are morphisms of schemes and is a sheaf isomorphism such that
X0
f0
Y0
b
a
/X
f
/Y
'
and : b∗ L −
→ L0 is an isomorphism of sheaves. This way we get a fibered
category p : Pol → Sch given by (X → Y, L) 7→ Y .
Proposition 8.12. Any fppf cover S 0 → S is of effective descent for Pol.
Proof. The proof was given as “self study”.
68
9
Example, torsors, principal homogeneous spaces
We start with an example. Then we define torsors and principal homogeneous spaces, and end with another example.
9.1
The moduli stack of curves of genus g ≥ 2
We define a category Mg that should parametrize families of curves of genus
g. We also show that it is in fact a stack.
f
Its objects are maps C −
→ S such that for each s ∈ S, the fiber Cs is a
geometrically connected, proper, smooth, curve of genus g.
The morphisms are just commutative diagrams
C0
/C
/S
S0
We get a functor p : Mg → Sch for free: It just sends a morphism C → S
to the codomian S. Clearly this makes Mg into a fibered category.
We have a morphism of fibered categories from Mg to Pol (the category
of polarized schemes) by sending C → S to the pair (C → S, Ω1C/S ).
For a fppf cover S 0 → S we have a commutative diagram comparing the
descent data for Mg and Pol:
Mg (S)
/ Pol(S)
'
/ Pol(S 0 → S)
Mg (S 0 → S)
The right arrow is an equivalence since every descent data for Pol is effective
(proven last lecture). A diagram stare shows that every object in Mg (S 0 →
S) lies in the essential image of Mg (S).
Also, the map Mg (S) → Mg (S 0 → S) is fully faithful by the effective
descent theorem of morphisms of sheaves on an arbitrary site, i.e. Theorem
8.1.
9.2
Torsors / principal homogeneous spaces
Let C be a site and µ a sheaf of groups on C. A µ-torsor on C is a sheaf P
on C together with a left action ρ : µ × P → P such that the following two
conditions hold:
69
T1. For all x ∈ C there exists a cover {Xi → X} such that P(Xi ) 6= ∅ for
all i.
T2. The map
µ×P →P ×P
(g, p) 7→ (p, gp)
is an isomorphism.
Exercise 10. The condition T2 is equivalent with: If P(X) 6= ∅, then the
action of µ(X) on P(X) is transitive.
♠
Solution 10. Let x, y ∈ P(X). Then by T2, the pair (x, y) ∈ P(X) × P(X)
is is equivalent to the pair (x, gx) for some g ∈ µ(X). But then y = gx, so
µ(X) acts transitively.
♥
A morphism of µ-torsors (P, p) → (P 0 , p0 ) is a morphism f : P → P 0
such that
µ×P
idµ ×f
/ µ × P0
p
P
f
p0
/ P0
Example 9.1 (Example of torsor). This example is from linear algebra. Let
T : V → W be a linear transformation and let w
~ ∈ W . Then the solution set
of T (w)
~ =w
~ is either empty or it is a torsor under the action of ker T . F
Now suppose that µ is represented by G, a flat finite type group scheme
(over X).
Definition 9.2. A principal G-bundle over X is a pair (π : P → X, ρ) where
π is a smooth surjective morphism of schemes, and ρ : G ×X P → P is a
map such that the following three conditions are true:
1. The following diagram commutes:
G ×X (G ×X P )
mg ×idP
idG ×ρ
/ G ×X P
ρ
ρ
G ×X P
70
/P
2. If e : X → G is the identity section, then
idP
P
/ G ×X P
e
e×idP
/% P
commutes.
3. There is an isomorphism p × pr2 : G ×X P → P ×X P .
A morphism (P, ρ) → (P 0 , ρ0 ) of principal G-bundles consists of an Xmorphism f : P → P 0 such that
G ×X P
ρ
idG ×f
P
/ G ×X P
f
ρ0
/ P0
commutes.
Now let (P, σ) be a principal G-bundle. We define a µ-torsor (P, ρ) as
follows: 9 The sheaf P is the sheaf represented by P , namely P = hP , with
action ρ induced by σ.
Condition T2 of the definition of a µ-torsor follows easily. It is the first
condition that need some big theorems: Since π : P → X is smooth and
surjective, there exists locally an étale section of X, by Corollary 3.9. This
implies in turn that there exists an fppf cover {Xi → X} with P(Xi ) 6= ∅
for all i. [[I don’t see how...]]
This discussion gives us a fully faithful functor ∗ from principal G-bundles
on X to µ-torsors on X, given by sending a pair (P, σ) to the functors they
represent: (hp , hσ ).
Proposition 9.3. If the structure map G → X is affine, then ∗ is an equivalence of categories.
Proof sketch. We need to show essential surjectivity (fully faithfullness is
free by Yoneda). That is, if (P, ρ) is a torsor, we want to see that P is
represented by a smooth X-scheme P .
Choose a fppf cover {Xi → X} with P(Xi ) 6= ∅ for all i. Then the
restriction of P to Xi is represented by an affine Xi -scheme Pi , namely
G ×X Xi (I don’t see why).
9
Recall that µ is represented by G.
71
We can glue this together to get a scheme P by descent for affine morphisms.
Here’s a long example:
Example 9.4. Let X be a scheme, and suppose that n is invertible on
X. This just means that n is a unit in O X (X)∗ . Then we define the group
scheme of roots of unity µn by µn (Y ) = {f ∈ O ∗Y | f n = 1}. We will consider
the category of µn -torsors on Xét .
Let Σn be the category defined as follows: Its objects are pairs (L, σ),
≈
where L is an invertible sheaf on X and σ : L⊗n −
→ O X is a trivialization
of the nth power of L.10 The morphisms (L, σ) → (L0 , σ 0 ) is a morphism of
invertible sheaves ρ : L → L0 such that
σ
L⊗n
ρ⊗n
/ OX
<
σ0
L0⊗n
Then we claim that we have an equivalence of categories F : Σn → Tors(µn )ét .
Given a pair (L, σ) ∈ Σn , let P(L,σ) be the étale sheaf associated to the
presheaf that maps a X-scheme U to the set of trivializations λ : O U → L|U
such that
λ⊗n / ⊗n σ /
OU
L|U
<O U
idOU
commutes. We have an action of µn (U ) on P(L,σ) (U ) as follows: given
ζ ∈ µn (U ), we send (ζ, λ) to ζλ, which makes sense because we have a
O X -module structure on L.
This makes P(L,σ) into a µn -torsor.
Now we see why we needed the étale topology: Given (L, σ) ∈ Σn , we
have no guarantee that P(L,σ) is non-empty in the Zariski-topology. However,
we can always find such a λ in the étale locally. [[Anyone sees why?]]
On the other hand, we have a natural functor G : Tors(µn )ét → Σn .
Given P a µn -torsor... (I don’t understand my notes...)
F
Here’s another example:
10
Compare with the discussion of root stacks in the last section.
72
Example 9.5. Let X = Spec k, where k is a field. In this case all line bundles
are trivial, hence Σn can be identified with a category having objects σ ∈ k ∗
and morphisms σ → σ 0 are just given by a λ ∈ k ∗ such that σ 0 = λn σ.
Then we get that isomorphism classes of σn -torsors on Spec két are in
bijection with k ∗ /(k ∗ )n , the nth roots of k.
F
73
10
Stacks and algebraic spaces
We are finally able to define a stack. Furthermore, we define algebraic spaces
and algebraic stacks.
10.1
Definition of stacks
We have finally arrived at the definition of stacks. Let C be a site.
Definition 10.1. A category fibered in groupoids p : F → C is a stack if for
all X ∈ C and coverings {Xi → X}i∈I , the functor
: F (X) → F ({Xi → X})
E 7→ (E, σcan )
is an equivalence of categories.
So, a stack is a fibered category in which the fibers can be defined locally,
i.e. by coverings.
Lemma 10.2. A category fibered in groupoids p : F → C is a stack if and
only if the following two conditions hold:
1. For all X ∈ C and x, y ∈ F (X), the presheaf Isom(x, y) on C/X is a
sheaf.
2. Any covering {Xi → X} is of effective descent.
Proof. Condition 1) says that the functor in the defintion is fully faithful,
and condition 2) says that it is essentially surjective.
Fibered categories only satisfying condition 1) are called prestacks.
Lemma 10.3. Let
F1
F2
d
c
/ F3
be a diagram of stacks fibered in groupoids over C. Then the fiber product
F1 ×F3 F2 is also a stack.
74
Proof hint. For any covering {Xi → X} in C, we have maps
(F1 ×F3 F2 )(X) → F1 (X) ×F3 (X) F2 (X)
and
(F1 ×F3 F2 )({Xi → X}) → F1 ({Xi → X}) ×F3 ({Xi →X}) F2 ({Xi → X}).
Both of these are equivalences of groupoids.
Given a category p : F → C fibered in groupoids, we can “stackify”.
Theorem 10.4 (Stackification). Let p : F → C be a category fibered in
groupoids. Then there exists a stack F a over C and a morphism of fibered
categories q : F → F a such that for all stacks G over C, the induced functor
HOMC (F a , G) → HOMC (F, G)
is an equivalence of categories.
Proof sketch. There are two main steps. In Step 1 one constructs F → F 0
that is universal among morphism to prestacks.
In Step 2 one constructs a morphism F 0 → F a that is universal for
morphisms to stacks.
The construction is essentially set-theoretical and consists of a lot of
book-keeping.
10.2
Algebraic spaces
Let S be a base-scheme and give SchS the étale topology (i.e. we are working
f
in the big étale site of S), and let F −
→ G be a morphism of sheaves.
Definition 10.5. We say that f is schematic or represented by schemes if
for all T ∈ SchS and morphisms T → G, the fiber product F ×G T is a
scheme.
Definition 10.6. Let P be a stable property of morphisms of schemes, that
is, it is stable under pullbacks. If f is represented by schemes, we say that
f has property P if for all S-schemes T , the morphism
pr2 : F ×G T → T
have the property P .
75
Exercise 11. If F, G are representable sheaves, then any morphism f : F →
G is represented by schemes.
♠
Solution 11. Let F = hX and G = hY , and let f : F → G be a morphism.
By the weak Yoneda lemma, this corresponds to a unique morphism f : X →
Y . Since the fiber product of schemes over schemes is a scheme, of course
F ×G T is a scheme. (this exercise was so trivial it wasn’t even fun)
♥
f
Exercise 12. Let X −
→ Y be a map of schemes and let P be some stable
property. Then f have the property P if and only if hX → hY have property
P.
♠
♥
Solution 12. This is equally trivial.
Lemma 10.7. Suppose ∆ : F → F × F is represented by schemes. Then
any morphism f : T → F where T is an S-scheme is represented by schemes.
Proof. Let T, T 0 be S-schemes, and consider the diagrams below:
T0
◦
T
f
/F
T × T0
◦
g
F
∆
f ×g
/F ×F
The fiber products in these two diagrams are isomorphic. To see this, just
write down the corresponding maps with elements.
Now we come to the definition of algebraic spaces:
Definition 10.8. An algebraic space over S is a functor X : Schop
S → Set
satisfying the following three conditions:
1. X is a sheaf (in the étale topology).
2. The diagonal mapping ∆ : X → X × X is represented by schemes.
3. There exists an S-scheme U and a surjective étale morphism U → X,
called an atlas of X.
Remark. In lieu of the last lemma, the third condition means the following:
for any S-scheme T and morphism T → X the morphism U ×X T → T is
étale and surjective. So in the way schemes are covered by affine schemes
in the Zariski topology, algebraic spaces are covered by schemes in the étale
topology.
76
10.3
Algebraic spaces as sheaf quotients
We’re still working in the étale topology.
Definition 10.9. Let X ∈ SchS . An étale equivalence relation on X is a
subscheme R ,→ X ×S X such that:
1. For all S-schemes T , the inclusion of sets
R(T ) ⊂ X(T ) ×S(T ) X(T )
is an equivalence relation (reflexive, symmetric, transitive).
2. The maps s, t : R → X induced by the projections are étale maps.
By taking quotiens of R we get a presheaf Schop
S → Set given by sending
an S-scheme T to the set-theoretic quotient X(T )/R(T ) = X(T )/ ∼. We
can sheafify to get a sheaf X/R on the site (SchS )ét .
Proposition 10.10. The construction above makes X/R into an algebraic
space.
Conversely, if Y is an algebraic space over S and X → Y is an étale
surjection where X is a scheme, then R := X ×Y X is a scheme, and the
inclusion R ,→ X ×S X is an étale equivalence relation. In addition X/R →
Y is an isomorphism.
Proof. We will only say a little bit about the proof.
For the first part, let Y = X/R. The hard part is showing that the
diagonal ∆ is representable, so we assume this has already been shown. We
will show that there exists a surjective étale atlas for Y . It is enough to
consider morphisms T → Y that factor through X (why?). Given such a
factorization, we get a commutative diagram
T ×Y X
/R
t
/X
T
s
/X
/Y
Since t by assumption is surjective étale, by base change, T ×Y X → T is
also surjective étale.
77
Finally, we say something about R = X ×Y X: It follows from the
commutative diagram
/ X ×S X
R
Y
∆/
Y ×S Y
that R is a scheme, since Y is an algebraic space.
10.4
Examples of algebraic spaces
Example 10.11. Let X be a scheme and let G be a finite group acting on X
via ρ : G×X → X. We say that the action is free if the map G×X → X ×X
given by (g, x) 7→ (x, ρ(g, x)) is injective.
Let X/G be the sheaf associated to the presheaf T 7→ X(T )/G. Then
X/G is an algebraic space.
F
Here is an example of an algebraic space that is not a scheme:
Example 10.12. This example comes from Hironaka, and is described in
Appendix B of [Har77]. We start with P3 with coordinates x0 , x1 , x2 , x3 .
And we define two curves by
C1 : x0 x1 + x1 x2 + x2 x0 = x3 = 0
C2 : x 0 x 1 + x 1 x 3 + x 3 x 0 = x 2 = 0
They intersect in two points, namely p1 = (1 : 0 : 0 : 0) and p2 = (0 : 1 : 0 :
0). Let i = 0, 1. On P3 \{pi }, blow up Ci and then blow up C1−i . We get
two blowups which we can glue along the inverse image of P3 \{p1 , p2 }. Call
the resulting scheme for Z.
There is an involution σ : Z → Z induced by x0 7→ x1 , x1 7→ x0 and
x2 7→ x3 and x3 7→ x2 . The automorphism σ have fix points, so we restrict to
the open subscheme Z 0 ⊂ Z where σ acts freely. Here we make the quotient
Z 0 /σ. By the above result, this is an algebraic space, but it is not a scheme.
The proof uses intersection theory.
F
10.5
More on algebraic spaces
Let P be a property of schemes stable in the étale topology. That is, such
that V ∈ SchS have P if and only if for every étale covering {Vi → V } all
the Vi ’s have P .
78
Definition 10.13. We say that an algebraic space X have a property P if
there is an étale surjection U → X such that U have P .
We say that a property P of morphisms of schemes is stable in the étale
topology if for any covering {Vi → V }, a morphism U → V have P if and
only if U ×V Vi → Vi have P for all i.
Definition 10.14. Suppose P is a property of morphisms of schemes that is
f
stable in the étale topology. Let X −
→ Y be a morphism of algebraic spaces
represented in schemes. Then we say that f has property P if there is an
étale surjection V → Y such that V ×X Y → V have property P .
In particular, we can talk about open and closed embeddings of algebraic
spaces.
Definition 10.15. We say that an algebraic space X is quasi-separated if
∆ : X → X × X is quasi-compact.
We say that a property P of morphisms of schemes is stable and local
f
ϕi
on the domain if for any scheme map U −
→ V and covering {Ui −→ U }, the
morphism f have P if and only if f ◦ ϕi have P for all i.
Definition 10.16. Let P be a propery of morphisms of schemes that is
f
stable and local on the domain in the étale topology. Let X −
→ Y be a
morphism of algebraic spaces. Then we say that f has P if there are étale
surjections v : V → Y and u : U → X such that the projection U ×Y V → V
have P .
So we can talk about morphisms of algebraic spaces that are étale, flat,
smooth, surjective, etc.
Exercise 13. Let f : X → Y be a representable morphism of algebraic
spaces and P a property that is local on the source. If f have P , then for
any étale surjection V → Y , the morphism X ×Y V → V have P .
♠
♥
Solution 13.
Exercise 14. If P is a property that is stable and local on the source, and
f
X −
→ Y is a morphism of algebraic spaces that have P , then if we have a
commutative diagram,
U
u
/ / X ×Y V
X
79
/V
f
/Y
where u is an étale surjection, then the composition U → V will have property P .
♠
♥
Solution 14.
Proposition 10.17. The full subcategory of Sch/S ét consisting of algebraic
spaces is closed under finite limits.
Proof. It is enough to show that it is closed under taking fiber products.
[[Long proof]]
A priori, an algebraic space X is only a sheaf in the étale topology. The
next lemma shows that it is a sheaf in the fppf topology.
Lemma 10.18. Let X be an algebraic space over S with quasi-compact diagonal ∆X . Then X is a sheaf in the fppf topology.
Exercise 15. Let C be a site and let X, R be sheaves on C. Let s × t :
R ,→ X × X be an inlusion such that for all U ∈ C, the inclusion R(U ) ⊆
X(U ) × X(U ) is an equivalence relation. Let X/R be the presheaf on C
associated to the presheaf U 7→ X(U )/R(U ). Then the diagram
R
t
s
/X
/ X/R
X
is cartesian.
♠
Solution 15. Easy exercise.
♥
The next exercise will be used in a later lecture:
Exercise 16. Let Y be an algebraic space over S. Let F be a sheaf on SchS
in the étale topology. Let g be a morphism g : F → Y . Show that if there
exists an étale surjection U → Y with U a scheme such that F ×Y U is an
algebraic space, then F is an algebraic space.
♠
♥
Solution 16.
80
11
Algebraic stacks
Now we have defined both stacks and algebraic spaces. Algebraic stacks are
something in between.
11.1
First results on algebraic stacks
Recall, that from now an, all stacks will be over SchS in the étale topology.
Definition 11.1. We say that a morphism of stacks f : X → Y is representable if for all S-schemes U and morphisms y : U → Y, the fiber product
X ×Y U is an algebraic space.11
f
Lemma 11.2. If X −
→ Y is a morphism of stacks, then for any algebraic
space V and morphism y : V → Y, the fiber product X ×Y V is an algebraic
space.
Proof. This should follow from Exercise 16.
Definition 11.3 (Algebraic stack). A stack X over S is algebraic if
1. The diagonal ∆ : X → X ×S X is representable and
2. there exists a smooth surjective morphism π : X → X where X is a
scheme.
An algebraic stack is often called an Artin stack.
Lemma 11.4. Let X/S be a stack. Then the diagonal ∆ is representable
if and only if for all S-schemes U and objects u1 , u2 ∈ X(U ), the sheaf
Isom(u1 , u2 ) on SchU is an algebraic space.
Furthermore, this happens if and only if for all algebraic spaces X and
maps x, y : X → X, the sheaf Isom(x, y) is an algebraic space.
Proof. Only a hint was given: The following diagram is cartesian.
Isom(u1 , u2 )
/U
X
11
∆
The gothic letter Y is math-speak for “Y”.
81
u1 ×u2
/X×X
Lemma 11.5. Suppose X/S is an Artin stack. For all diagrams
X
Y
x
/X
y
where X, Y are algebraic spaces, the fiber product X ×X Y is an algebraic
space.
In particular, every morphism X → X where X is an algebraic space is
representable.
“Proof ”. Observe that X ×X Y ' Isom(pr∗1 x, pr∗2 y) over X ×S Y .
Here’s a lengthy example:
Example 11.6. Let X be an algebraic space. Let G/S be a smooth group
scheme acting on X. We can define a stack [X/G] as follows: The objects
are triples (T, P, π), where:
1. T is a scheme over S.
2. P is a GT = G ×S T -torsor on the big étale site of T .
3. π is a morphism π : P → X ×S T = XT that is a GT -equivariant
morphism of sheaves on SchT .
The morphisms (T 0 , P 0 , π 0 ) → (T, P, π) are pairs (f, f b ) where f : T 0 → T is
'
a map of S-schemes and f b : P 0 −
→ f ∗ P is an isomorphism of GT -torsors on
SchT such that the diagram
P0
f∗
/ f ∗P 0
π
#
/P
f ∗π
X ×S T 0
id×f
π
/ XT
commutes. The dotted arrows automatically commute.
Now descent for sheaves on sites shows that [X/G] is a stack. It is actually
an Artin stack: We first have to check if ∆ is representable. Let T ∈ SchS
and let (Pi , πi ) be objects in [X/G] over T (we’re omitting the T from the
notation here since we’re looking over the fibers anyway).
To show that I := Isom((P1 , π1 ), (P2 , π2 )) is an algebraic space, we can
replace T by an étale cover (this was an exercise from last time). Thus we
82
can assume that P1 and P2 are trivial torsors. Fix isomorphisms σi : Pi →
GT = G ×S T such that we can identify the π in the definition with a map
πi : GT → XT . Now the sheaf I can be identified with the sheaf that to T 0
over T associates the set of elements g ∈ G(T 0 ) such that the diagram
GT 0
mg
π1
/ GT 0
π2
" XT
commutes (the upper arrow is just multiplication by g). Now, since we are
dealing with group schemes, commutativity is equivalent with π1 (e) = π2 (g),
where e ∈ GT (T ) is the identity section.
Thus we can identify I with the fiber product of the diagram:
/ GT
I
XT
∆/
XT ×T XT
This implies that I is a scheme (and in particular, an algebraic space), by
representability of the diagonal morphism, and it follows by Lemma 11.4
that the diagonal morphism of [X/G] is representable.
F
Exercise 17. Find a smooth surjection onto [X/G].
♠
Solution 17.
♥
Definition 11.7. The classifying stack of G is the stack quotient [S/G], and
is denoted by BG.
Remark. If we had come so far as to define dimension of stacks, we would
have seen that the dimension of [S/G] is dim S − dim G, so we see that
arbitrary large negative dimensions of stacks occur (!!).
Lemma 11.8. Artin stacks have fiber products. That is, if X1 , X2 , X are
Artin stacks, then the fiber product X1 ×X X2 is also an Artin stack.
The proof is not difficult but very long, so we skip it.
Definition 11.9 (Inertia stacks). Let X/S be an Artin stack. The inertia
stack (norwegian: treghetsstacken) IX of X is the fiber product X ×X×S X
X.
83
Explicitly, the objects are pairs (x, g) where x ∈ X(T ) for some scheme
T and g is an automorphism of x ∈ X(T ). The morphisms (x0 , g 0 ) → (x, g)
consists of a morphism f : x0 → x in X such that
f
x0
/x
' g0
' g
x0
/ x
f
commutes. The map IX → X sends (x, g) → x.
Example 11.10. Let X = [X/G]. Then IX is the stabilizer of G acting on
X.
F
11.2
Properties of Artin stacks
Let P be a property of S-schemes that is stable in the smooth topology.
Definition 11.11. We say that an Artin stack X/S has property P if there
is a surjective smooth map π : X → X where X is a scheme that have the
property P .
Lemma 11.12. Let P be a property as above and X/S an Artin stack with
P . Then for any morphism y : Y → X where Y is an algebraic space, Y also
have P .
Proof. The proof was given as an “exercise with hint”. Consider the following
diagram:
/X
Y ×X X
Y
π
/X
y
It follows (why?) that the left vertical arrow is also smooth and surjective.
Let X and Y be maps of Artin stacks. A map12 for f is a commutative
diagram
h
g
X
q
p0
X
12
/' Y
/ X0
f
(4)
p
/Y
Not “map” as in “function”, but “map” as in “atlas”.
84
where g and p are smooth og surjective and X and Y are algebraic spaces.
If X, Y are schemes, we say that it is a map of schemes.13
Definition 11.13. Let P be a property of morphisms between schemes
that is stable and local on the source in the smooth topology (for example
smooth maps, locally of finite presentation, etc.). We say that a morphism
f : X → Y has the property P if there is a map for f of schemes where h
has the property P .
Having a property P is independent of the map:
f
Proposition 11.14. Let P be as above. Then the map X −
→ Y have P if
and only if for all maps for f , h has P .
Proof. Fix one map as in Equation 4, and let Y 0 → Y be a smooth surjection
of algebraic spaces. We get a commutative diagram
h0
X ×Y Y 0
g0
/ X0
g
X
+/
/ X ×Y Y 0
Y0
/Y
f
X
/Y
Consider the “outer” diagram:
X ×Y Y 0
/ X ×Y Y 0
/Y0
/Y
X
This is also a map for f . Thus we see that the morphism h in Equation 4
have property P if and only if f 0 : X ×Y Y 0 → Y 0 have P (the projection
X ×Y Y 0 → X is smooth and surjective).
Now fix two maps (i = 1, 2):
hi
Xi
gi
q
pi 0
X
13
/' Yi
/ X0
This can’t be right? It sounds terrible.
85
f
pi
/Y
We want to show that h1 have P if and only if h2 have P . Let Y 0 = Y1 ×Y Y2 .
By the above discussion, and by considering the maps pj : Y 0 → Yj for
j = 1, 2, one sees that it is enough to consider the case where Y1 = Y2 and
p1 = p2 . So write X0 for the fiber product X ×Y Y1 = X ×Y Y2 . Then we
have a commutative diagram, where X := X1 ×X X2 :
h2
X
pr2
/ X2
pr1
g2
/ X0
>
/) Y
_
g1
h1
X1
X
/Y
Now since P is local on the source in the smooth topology, the morphism h1
will have P if and only if h1 ◦ pr1 : X → Y have P , but this happens if and
only h2 ◦ pr2 have P and this happens if and only if h2 have P .
Definition 11.15. Let P be a property of morphisms of algebraic spaces
that is stable in the smooth topology on the category of algebraic spaces
over S.
We say that a representable morphism of Artin stacks f : X → Y have P
if for alle morphisms Y → Y where Y is an algebraic space, the morphism
X ×Y Y → Y have P
Thus it makes sense to say that representable morphisms are étale, smooth,
separated, proper, etc.
Definition 11.16. An Artin Stack is a Deligne-Mumford stack if there is
an representable étale surjection X → X where X is a scheme.
We abbreviate “Deligne-Mumford stack” by just saying “DM stack”.
Recall that a morphism of scheme g : Z → W is formally unramified
if for all closed embedding S0 ,→ S of affine schemes defined by nilpotent
ideals, the map
Z(S) → W (S) ×W (S0 ) Z(S0 )
is an injection, and this in turn is equivalent to Ω1Z/W = 0. Also, being
formally unramified is stable and local on the source in both the étale and
the smooth topology. Thus it makes sense to say that maps between Artin
stacks are formally unramified.
86
Theorem 11.17. Let X/S be an Artin stack. Then X/S is DM if and only
if the diagonal ∆ : X → X ×S X is formally unramified.
This is a big theorem and we won’t prove it. In some sense it says
this: X is DM if and only if none of the objects of X allow infinetesimal
automorphisms. We will try to explain what this means in the rest of the
lecture.
Here’s a fact: The diagonal ∆ is formally unramified if and only if for all
algebraically closed field k and objects x ∈ X(k), the automorphism group
Autx is a finite, reduced k-group scheme.
It fits into a cartesian diagram (? why??)
/ Spec k
Autx
X
∆
/X×X
[[ the lecturer goes on to prove the converse but I don’t understand it ]]
11.3
The moduli stack of curves of genus ≥ 2
Recall the fibered category Mg from Subsection 9.1.
Theorem 11.18. The stack Mg is a Deligne-Mumford stack.
This is a hard theorem and proven by Deligne and Mumford in the 70’s.
f
Lemma 11.19. Let (S, C −
→ S) be an object in Mg and LC/S the invertible
⊗3
sheaf Ω1C/S . Then:
1. The pullback f∗ LC/S is a locally free sheaf of rank 5g − 5 on S.
2. The map (“the counit of adjunction”) f ∗ f∗ LC/S → LC/S is surjective,
and the associated map C → P(f∗ LC/S ) is a closed embedding.
3. For any morphism g 0 : S 0 → S, the map
g ∗ f∗ LC/S → f∗0 g 0 LC/S
is an isomorphism.
Proof. For 1) and 2) we can assume that S = Spec k and that k = k̄,
by flat base change [Har77, Chapter III, §12, Thm 12.11]. We claim that
dimk H 0 (C, LC/S ) = 5g − 5 and that H i (C, LC/S ) = 0 for all i > 0. The last
87
statement follows by Serre duality (I don’t see how?). For the first part, note
that the degree of the canonical divisor on C is 2g − 2 by Example 1.3.3 in
Chapter V in Harthorne. Hence dimk H 0 (C, LC/S ) = 2 deg(Ω1C/S ) + 1 − g =
3(2g − 2) + 1 − g = 5g − 5.
For 2), somehow use that k = k̄ and that LC/S is very ample.
∼
Let Mg be the following functor on Sch: For each S-scheme it associates
f
'
→ f∗ LC/S ). What
the set of isomorphism classes of pairs (C −
→ S, σ : O S5g−5 −
f0
is an isomorphism of such pairs? Let (C 0 −→ S 0 , σ 0 : O 5g−5
→ f∗0 LC 0 /S )
S
be another such pair. Then an isomorphism is an isomorpism of curves
α : C 0 → C such that the following diagram commutes:
O 5g−5
S
σ0
/ f 0 LC 0 /S
∗
σ
$
f∗ LC/S
∼
We have that G := GL5g−5 acts on Mg : On an S-point it works like this:
g · (C → S, σ) = (C → S, σ ◦ g) for g ∈ G(S).
∼
We have a map π : Mg → Mg defined by sending a pair (C/S, σ) to the
pair (S, C).
∼
It turns out that we have an isomorphism [Mg / GL5g−5 ] ' Mg . In
particular, Mg is an Artin stack!
To show that the diagonal ∆ : Mg → Mg × Mg is formally unramified,
we use the isomorphism. It is enough to show that for any algebraically
closed field k and smooth genus g curve C/k, the k-group scheme Autk (C)
is reduced. The proof uses that g ≥ 2.
88
12
Root stacks and the Weil conjecures
In this last lecture we give another example of a stack, and in the last part
we prove state the Weil conjectures and define a Weil cohomology theory.
12.1
Root stacks
Let X be a scheme. Recall that a Cartier divisor on X is given by global
sections fi ∈ Γ(Ui , K∗ ) for a (Zariski) covering {Ui }, such that fi /fj ∈
Γ(Ui ∩ Uj , O∗ ), where K is the sheaf of total quotient rings on O. A Cartier
divisor is effective if already all the fi ’s are global sections of Γ(Ui , O∗ ).
Stating things more compactly, giving a Cartier divisor is the same as giving
a global section of K∗ / O ∗ .
Note that effective Cartier divisors are in 1-1 correspondence with locally
principal closed subschemes of X.
Let D be a given effective Cartier-divisor on X and fix a natural number
n. Then one can ask: does there exist an effective Cartier-divisor E on X
such that D = nE?14
More generally, one can ask: does there exists a scheme Y and a map
f : Y → X and a Cartier divisor E on Y such that nE = f ∗ D? The problem
is that divisors doesn’t always pull back! For example, for curves, divisors
only pull back for finite maps! (see [Har77], Chapter II, Proposition 6.8).
So, as one does in mathematics, when something doesn’t work, we generalize!
Definition 12.1. A generalized effective Cartier-divisor (geCd ) on X is
a couple (L, ρ) where L is an invertible sheaf on X and ρ is a morphism
L → OX of O X -modules. A morphism of couples (L0 , ρ0 ) → (L, ρ) is a
commutative diagram:
/L
L0
ρ0
!
OX
~
ρ
Here are some examples.
Example 12.2. Let D ⊆ X be a locally principal subscheme of X and let
ID be the associated ideal sheaf. Then the inclusion i : ID ,→ O X makes
(ID , i) a geCd.
F
14
Recall that given a Cartier divisor D, one can form an associated invertible sheaf
L(D), and we have a group operation by E + E 0 ↔ L(E) ⊗ L(E 0 ).
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Example 12.3. Any invertible sheaf L give rise to a geCd given by (L, 0),
where 0 is the zero map 0 : L → O X .
F
We can multiply geCd’s:
Definition 12.4. Given two geCd’s, (L, ρ) and (L0 , ρ0 ), we define their product to be
(L, ρ) · (L0 , ρ0 ) := (L ⊗ L0 , ρ ⊗ ρ0 ).
Here ρ ⊗ ρ0 is the morphism L ⊗ L0 → O X ⊗ O X ≈ O X .
But behold! Now we can pull back divisors! For let g : Y → X be some
scheme map, and let (L, ρ) be a geCd on X. This gives
(g ∗ L, g ∗ ρ : g ∗ L → g ∗ O X = O Y ),
a geCd on Y .
We want to define a fibered category D. Its objects are pairs (T, (L, ρ))
where T is a scheme and (L, ρ) is a geCd. A morphism in D consists of a
≈
scheme map g : T → T 0 and an isomorphism of geCd’s (L0 , ρ0 ) −
→ (g ∗ L, g ∗ ρ).
Via the forgetful functor p : D → Sch given by (T, (L, ρ)) 7→ T , we get a
category over Sch, where the fibers over X are the category of geCd’s on X.
If we work in the étale topology (or the fppf topology), then it follows that
D is a stack by descent for invertible sheaves.
Proposition 12.5. We have an isomorphism of stacks
D ≈ [A1 /Gm ]
In particular, D is an algebraic stack.
Sketch of proof. We consider the prestack {A1 /Gm } whose objects are pairs
(T, f ∈ Γ(T, O T )), where T is a scheme, and a morphism to (T 0 , f 0 ∈
Γ(T 0 , O T 0 )) is given by a pair (g, u) where g : T 0 → T og u ∈ Γ(T 0 , O ∗T 0 )
such that f 0 = u · g # (f ) in Γ(T 0 , O T 0 ).15
We have a morphism of fibered categories {A1 /Gm } → D given by sending an object (T, f ) → (T, (O T , ·f )) and a morphism (g, u) to the morphism
(T 0 , (O T 0 , ·f 0 )) → (T, (O T , ·f )) given by multiplication by f .
One can show that this induces an equivalence of stacks after stackifying
(but I don’t have a reference for this).
Recall that A1 represents the functor T 7→ Γ(T, O T ) and Gm represents the functor
T 7→ Γ(T, O ∗T )
15
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Now we consider the morphism A1 → A1 given by t mapstotn and the
morphism Gm → Gm given by u 7→ un . This induces an endomorphism
ρn : [A1 /Gm ] . This endomorphism corresponds to taking nth powers of
sheaves:
(T, (L, ρ)) 7→ (T, (L⊗n , ρ⊗n )).
Now fix a geCd (L, ρ) on X and a natural number n. Let Xn be the
f
fibered category over Sch with objects triples (T −
→ Xk, (M, λ), σ) where
≈
⊗n
∗
∗
(M, λ) is a geCd on T and σ : (M , λ) −
→ (f L, f ρ) is an isomorphism of
invertible sheaves.
f0
f
A morphism (T 0 −→ X 0 , (M 0 , λ0 ), σ 0 ) → (T −
→ X, (M, λ), σ) is given by
≈
a pair (h, hb ) where h : T 0 → T is an X-morphism, and hb : (M 0 , λ0 −
→
(h∗ M, h∗ λ) is an isomorphism of geCd’s. In addition, we demand that the
diagram
hb⊗n
M 0,⊗n
'
λ0
f 0∗ L
≈
/ h∗ M ⊗n
h∗ λ
w
h∗ f ∗ L
commutes.
Definition 12.6. The nth root stack of (L, ρ) is the stack Xn defined above.
That Xn actually is a stack follows from the next theorem:
Theorem 12.7. The following holds:
1. Xn is an algebraic stack.
2. If L = O X and ρ is given by f ∈ Γ(X, O X ), then Xn is isomorphic
with the quotient stack of SpecX (O X [T ]/(T n − f )) by the action of µn
given by ζ · T = ζT .16
3. If n is invertible on X, then Xn is a DM-sheaf.
Proof. We only prove number 1). This follows by noting that Xn is the fiber
product of the diagram
/D
Xn
D
(L,ρ)
The other parts are harder.
16
µn is the group of unit roots.
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ρn
/D
12.2
The Weil conjectures
All that was said in this lecture can be found in the Appendix of [Har77],
except the definition of a Weil cohomology theory, which we state here for
completeness.
A Weil cohomology theory for schemes that are reduces and of finite type
over a field k with coefficients in a field K with charK = 0 is a contravariant
functor from such schemes to graded commutative K-algebras. We write
X 7→ H ∗ (X) = ⊕H i (X).
This graded K-algebra should also have a cup product.
We also have a trace map TrX : H 2 dim X (X) → K, and for every closed
subscheme Z ,→ X a class c(Z) ∈ H 2c (X), where c = codimZ. This should
satisfy the following list of axioms:
A1. Finite-dimensionality. Every H i (X) should be a finite-dimensional Kvector space. And furthermore, H i (X) = 0 for i < 0 and i > 2 dim X.
A2. Künneth formula. We have an isomorphism
H ∗ (X) ⊗K H ∗ (Y ) → H ∗ (X × Y )
α ⊗ β 7→ pr∗X α ∪ pr∗Y β
A3. Poincaré-duality. The trace map TrX is an isomorphism, and for 0 ≤ i ≤
2 dim X, the K-linear map
H i (X) ⊗K H 2 dim X−i → K
α ⊗ β 7→ α ∪ β
is a perfect pairing of vector spaces.
A4. Multiplicativity of the trace map. For α ∈ H 2 dim X (X) and β ∈ H 2 dim Y (Y ),
we have
TrX×Y (pr∗X α ∪ pr∗Y β) = TrX (α)TrY (β).
A5. Yoneda product. For closed subschemes Z ,→ X and W ,→ Y , we have
c(Z × W ) = pr∗X (c(Z)) ∪ pr∗Y (c(W )).
A6. Pushforward. Given a map f : X → Y , there is map f∗ : H ∗ (X) →
H ∗ (Y ).
A7. Pullback. Given a map f : X → Y , there is a map f ∗ : H ∗ (Y ) → H ∗ (X).
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A8. Normalization. For X = Spec k, we have c(X) = 1 and TrX (1) = 1.
For more on Weil cohomology theories, the reader is adviced to do a
Google search.
We concluded the course with two theorems:
Theorem 12.8 (Lefschetz’ trace formula). Let ϕ : X → X be smooth,
projective of dimension N . Then
X
∆ · Γϕ =
(−1)n Tr(ϕ∗ |H n (X))
If ∆ and Γϕ intersects transversely (meaning that any irreducible component
of ∆ ∩ Γϕ have codimension codim ∆ + codim Γϕ ), then
∆ · Γϕ = #{x ∈ X|ϕ(x) = x}.
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A
Exam
Here are the five topics to be talked about on the exam:
• Moduli problems. See Section 1. Examples includes the Grassmannian (subsection 4.2 and onwards), genus g curves (subsection 9.1 and
subsection 11.3.
• Vektor bundles. See Section 1, subsection 1.4. Also see the book by
Frank Neumann [Neu09], especially the first chapter.
• Faithfully flat descent. This is in Section 8.
• Fibered categories. This is Section 6. Also see examples.
• The moduli stack of curves of genus ≥ 2, Mg . Subsections 9.1 and
11.3.
B
References to the litterature
Stacks are still so obscure that there are no canonical literature unless one
wants to read the french EGA/SGA.
Much of the problem with studying stacks is that they are so abstract
that intuition and motivation for the concepts are hard to grasp.
There are several good articles and sections in books that tries to explain
and give motivation for the concepts introduced here. For a really good one,
see “Picard groups of moduli problems” [Mum65] by David Mumford, in
which he motivates the introduction of other topologies, and uses them to
compute the Picard group of Mg .
In Hartshorne [Har10], Remark 27.7.1, there are two pages explaining
why stacks are needed.
There is also the note by Barbara Fantechi [Fan01], “Stacks for Everybody”, using concrete examples. A more technical, but more motivating
article is “Algebraic stacks” by Tomás Gómez [Góm01].
Of course there’s also the “Stacks Project” [Sta13], but unless you’re
already an expert in the field, most of the stuff is unreadable.
B.1
Background litterature
For background in algebraic geometry, there is of course [Har77], but also
sometimes [Har10]. Milne has a lot of stuff on properties of étale and flat
maps, and also a lot on sheaves, see [Mil80].
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The last chapter of [EH00] has a lot to say about the functorial point of
view in algebraic geometry.
References
[EH00] David Eisenbud and Joe Harris. The geometry of schemes, volume
197 of Graduate Texts in Mathematics. Springer-Verlag, New York,
2000.
[Eis95] David Eisenbud. Commutative algebra, volume 150 of Graduate
Texts in Mathematics. Springer-Verlag, New York, 1995. With a
view toward algebraic geometry.
[Fan01] Barbara Fantechi. Stacks for everybody. In European Congress of
Mathematics, Vol. I (Barcelona, 2000), volume 201 of Progr. Math.,
pages 349–359. Birkhäuser, Basel, 2001.
[Góm01] Tomás L. Gómez. Algebraic stacks. Proc. Indian Acad. Sci. Math.
Sci., 111(1):1–31, 2001.
[Har77] Robin Hartshorne. Algebraic geometry. Springer-Verlag, New York,
1977. Graduate Texts in Mathematics, No. 52.
[Har10] Robin Hartshorne. Deformation theory, volume 257 of Graduate
Texts in Mathematics. Springer, New York, 2010.
[Kle69] Steven L. Kleiman. Geometry on Grassmannians and applications
to splitting bundles and smoothing cycles. Inst. Hautes Études Sci.
Publ. Math., ?(36):281–297, 1969.
[Mey13] Fredrik Meyer. Degenerations of G(3,6). Master’s thesis, University
of Oslo, 2013.
[Mil80] James S. Milne. Étale cohomology, volume 33 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980.
[Mum65] David Mumford. Picard groups of moduli problems. In Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963), pages
33–81. Harper & Row, New York, 1965.
[Neu09] Frank Neumann. Algebraic stacks and moduli of vector bundles.
Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA),
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Rio de Janeiro, 2009. 27o Colóquio Brasileiro de Matemática. [27th
Brazilian Mathematics Colloquium].
[Nit05] Nitin Nitsure. Construction of Hilbert and Quot schemes. In Fundamental algebraic geometry, volume 123 of Math. Surveys Monogr.,
pages 105–137. Amer. Math. Soc., Providence, RI, 2005.
[Ros94] Jonathan Rosenberg. Algebraic K-theory and its applications, volume 147 of Graduate Texts in Mathematics. Springer-Verlag, New
York, 1994.
[Sta13] The Stacks Project Authors. stacks project. http: // stacks. math.
columbia. edu , 2013.
[Vak13] Ravi Vakil. Math 216: Foundations of algebraic geometry. http:
// math. stanford. edu/ ~vakil/ 216blog/ , juni 2013.
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